Quantized Repetitions of the Cuprate Pseudogap Line
QQuantized Repetitions of the Cuprate Pseudogap Line
Vincent Sacksteder IV ∗ Department of Physics, Royal Holloway University of London,Egham Hill, Egham, TW20 0EX, United Kingdom
The cuprate superconductors display several characteristic temperatures which decrease as thematerial composition is doped, tracing lines across the temperature-doping phase diagram. Foremostamong these is the pseudogap transition. At a higher temperature a peak is seen in the magneticsusceptibility, and changes in symmetry and in transport are seen at other characteristic temper-atures. We report a meta-analysis of all measurements of characteristic temperatures well above T c in strontium doped lanthanum cuprate (LSCO) and oxygen doped YBCO. The experimentalcorpus shows that the pseudogap line is one of a family of four straight lines which stretches acrossthe phase diagram from low to high doping, and from T c up to 700 K. These lines all originatefrom a single point near the overdoped limit of the superconducting phase and increase as dopingis reduced. The slope of the pseudogap lines is quantized, with the second, third, and fourth lineshaving slopes that are respectively 1 / , / , and 1 / PACS numbers: 74.72.Gh,74.72.Kf,74.25.Dw
The cuprate high T c superconductors are built fromcopper oxide planes, and when these planes are dopedwith holes superconductivity occurs. Temperaturesabove the superconducting critical temperature T c makethe cuprates effectively two dimensional by disruptingthe weak interplane coupling. Nonetheless the cupratesexhibit a wide variety of mysterious features at tem-peratures far exceeding T c . One of the first to be no-ticed is a broad peak in the magnetic susceptibility ata temperature T max , which is accompanied by qualita-tive changes in electrical and thermal transport. Atlow dopings T max reaches temperatures as high as 700K, and it decreases linearly as the parent compound isdoped with holes. Focus has since shifted toward a pseu-dogap temperature T ∗ which is is substantially smallerthan T max and again decreases linearly with hole dop-ing. T ∗ marks a depletion in the density of states, andhas been measured by many experimental techniques in-cluding angularly resolved photoemission spectroscopy(ARPES). Over the years a perplexing cornucopiaof other doping-dependent signals have been identified,including a lower pseudogap temperature T ν , charge den-sity waves, nematic order, and a linear resistance ex-tending to very high temperatures marking the ”strangemetal” regime. In this paper we identify an unexpected regularityin the cuprate phase diagram: four of the experimen-tally measured doping-dependent temperatures, includ-ing T max , the pseudogap temperature T ∗ , and the lowerpseudogap line T ν , belong to a family which radiates froma common intersection and has slopes determined by aquantization rule. Our results are based on a compre-hensive and exhaustive survey of research papers whichmeasure temperature dependence and report numericalvalues of characteristic temperatures well above T c . We report only characteristic temperatures at which clearlyidentifiable signals occur, for example peaks or kinks inthe temperature dependence, or extinction of a diffrac-tion peak at a particular temperature. In particular, thereported temperatures have been realized experimentally,rather than being derived by extrapolation from lowertemperatures.In order to remove any confusion about material-specific aspects of the phase diagram, we have sepa-rated data obtained from different compounds and dop-ing techniques. We first report in Section I the charac-teristic temperatures of strontium-doped La − x Sr x CuO (LSCO). Separately, in Section II we report the char-acteristic temperatures of oxygen-doped YBa Cu O δ (YBCO). These two compounds and doping strate-gies, plus a third compound Bi Sr Ca n − Cu n O n +4+ δ (BSCCO), have been used for the large majority of dataon characteristic temperatures of cuprates well above T c ;our restriction to LSCO and YBCO is not terribly selec-tive. In the following sections we compare the LSCO andYBCO data and draw conclusions about which aspectsof the high T c phase diagram are universal and which arematerial specific.In gathering data we have followed standard prac-tice among reviews of the high T c phase diagram andthe pseudogap by focusing on those experimental sig-nals which are understood to reflect the carriers thatmediate superconductivity, and omitting the extensiveexperimental literature that focuses on ionic behavior.Along these lines we omit thermal history dependenceand hysteresis, oxygen movement and ordering, andmechanically-oriented observables such as lattice con-stants and internal friction.Within these restrictions we have made every effortto be comprehensive. Our goal has been to gather into a r X i v : . [ c ond - m a t . d i s - nn ] J un one place all experimental data on characteristic temper-atures above T c .We separate the results of distinct experimental groupsinto distinct data sets. When a group used several exper-imental techniques, identified several characteristic tem-peratures in their data, or re-analyzed data from otherpublications, we separate these distinct results into dis-tinct data sets. When a group republished or revisedtheir results, we use the latest results. Full details abouteach data set and about which papers did and did notmatch our selection criteria are included in the appen-dices. I. LSCO DATA
Figure 1 summarizes the experimental corpus onstrontium-doped LSCO. While this compound has a rel-atively low maximum T c of about 39 K, it is stable upto 1000 Kelvin and the full range of dopings across thecuprate superconducting dome can be explored. The cor-pus contains a total of twenty distinct data sets fromthirteen distinct experimental groups published between1990 to 2018, using a variety of sample preparationtechniques and experimental probes developed over thedecades. Five more data sets exist on characteristictemperatures in neodymium and europium-doped LSCO,with excellent agreement with the strontium doped re-sults shown here. Appendix B gives full details on eachof these data sets and on three more strontium-dopeddata sets which we have omitted on a case by case ba-sis. In order to show the slope and profile of individualdata sets we plot lines connecting the data points withineach set.Figure 1 shows that the data sets are in remarkableagreement, forming a pattern of four distinct character-istic temperatures which each start high at small dopingand decrease as doping is increased. Despite the varietyof samples and probes, the scatter in data points aroundeach characteristic temperature is small, of order ±
15 Kon the lowest three characteristic temperatures (orange,green, and blue lines), and somewhat larger on the high-est characteristic temperature (red lines). Gaps whichare much larger than this scatter separate each of thecharacteristic temperatures, allowing each data set to beunambiguously assigned to one of the four characteristictemperatures. We number characteristic temperaturesfrom the top, with n = 1 on the highest (red) line, n = 2on the next highest (orange), etc.The topmost red lines (four data sets) show T max ,visible both as a peak in the magnetic susceptibilityand as kinks in the resistance and thermoelectric power(TEP). The yellow lines (second from the top,five data sets) show the temperature T LT O of the well-known transition from the high symmetry tetragonalphase to a lower symmetry orthorhombic phase, whichhas generally though not unanimously been regardedas simply a structural phase transition with little re- T [ K ] T max n =1 T LTO n =2 T ∗ n =3 T ν n =4 T c T Neel
FIG. 1: (Color online.) The family of pseudogap lines inLSCO. The black guides to the eye have slopes proportionalto 1 /n and meet at doping p = 0 .
26. Individual pseudogapdata sets, each from a distinct experimental group measuringa distinct experimental probe and signature, are shown asfilled circles connected by lines. The hallmark of the first (red)pseudogap line T max is a peak in the magnetic susceptibility. C symmetry is broken on the second (orange) line T LTO , andthe ARPES density of states develops a pseudogap on thethird (green) line T ∗ . The fourth pseudogap line T ν (blue) ismarked by transport signatures. Purple lines show NMR, Hallangle, heat capacity, and neutron scattering data sets. Thepink line shows the superconducting T c . An antiferromagneticphase (grey line) is found at low dopings. lation to high T c . This transition manifestsclearly in neutron scattering, X-ray diffraction, resis-tance, and TEP measurements. The green lines (thirdfrom the top, four data sets) mark the pseudogap temper-ature T ∗ , where ARPES shows that a pseudogap opensin the density of states. Like T max and T LT O , T ∗ isseen also in the resistance and the TEP. . Theblue lines (fourth from the top, three data sets) are arecently identified second pseudogap transition T ν visi-ble in the Nernst effect and in nuclear magnetic reso-nance (NMR). This transition is well attestedby four additional Nernst and resistance data sets in neodymium-doped and europium-doped variants ofLSCO, which give temperatures that agree very well withthe strontium-doped LSCO data.
We include alsofour NMR, Hall angle, heat capacity, and neutron scat-tering data sets (in purple) which suggest similar lines atlower temperatures.
The fact that over the last thirty years experiments onLSCO’s temperature dependence consistently have founddistinctive behavior (peaks, kinks, extinction of diffrac-tion peaks, etc.) on these four fairly crisp lines, and notin the intervening gaps, indicates that this pattern haspredictive power. We expect that future experiments willcontinue to find smooth temperature dependence in the T [ K ] n =1 n =2 n =3 n =4 FIG. 2: (Color online.) Intersection of the pseudogap familyin LSCO near T = 0 , p = 0 .
26. The straight lines showlinear fits to individual pseudogap data sets, displayed as filledcircles. gaps and anomalous behaviors on these lines. This ob-served pattern and prediction are our most importantresults.The four characteristic temperatures just discussed fol-low a pattern which we have highlighted with thin blacklines.
Both the experimental data and the black linesare organized as harmonics of the uppermost line, withthe n -th line having a slope equal to 1 /n times that ofthe highest n = 1 line. Moreover all four lines radi-ate from a point p ≈ .
26 that lies near the high-dopinglimit p c = 0 . of the superconducting dome. Fig-ure 2 displays linear regressions of the eighteen pseudo-gap data sets which contain more than one data point.The linear regressions of fourteen out of eighteen pseu-dogap data sets intercept the T = 0 axis in the interval p = [0 . , . . , . T = 0 intercept on ornear the high-doping edge of LSCO’s superconductingdome, and clearly exclude the possibility suggested bysome that the intercept lies well inside the dome, near p = 0 . Both the agreement of intercepts and the pattern of1 /n slopes are remarkable given that we have used verybroad selection criteria embracing an extremely diverseset of experimental techniques and sample preparationprotocols, publications from 1990 to the present, and sig-nificant scatter within individual data sets. While it isoften suggested that the pseudogap temperature is sensi-tive to the experimental technique, we find instead thatall published measurements agree on the same simplepattern.These results force the conclusion that all four char-acteristic temperatures are members of the same fam-ily. This family pervades most if not all of the cuprate T [ K ] n =1 T ∗ n =2 T nem n =3 T ∗ ∗ n =4 T Neel T c FIG. 3: (Color online.) The family of pseudogap lines inYBCO. The black guides to the eye have slopes proportionalto 1 /n and meet at doping p = 0 . n = 1 pseudo-gap line a transport anomaly (red data set) occurs. C andintra unit cell symmetries are broken on the second (orange)line T ∗ , while on the third (green) line T nem transport be-comes nematic and time reversal symmetry is broken. Thefourth pseudogap line T ∗∗ (blue) is marked by transport sig-natures. Purple lines show additional transport data sets.The pink line shows the superconducting T c . Underneath T c three-dimensional charge density wave order (dashed grey)occurs, and an antiferromagnetic phase (solid grey) is foundat low dopings. phase diagram: in LSCO the n = 2 pseudogap line, i.e.the transition to orthorhombic symmetry, persists to zerodoping and 515 K, and the n = 1 pseudogap line extendsto above 700 K. Since the n = 1 , n = 3 , and n = 4 linesin this family are not structural transitions, the tetrag-onal to orthorhombic structural symmetry breaking onthe n = 2 line must be a subsidiary signal of underlyingelectronic nematic order. II. YBCO DATA
Figure 3 summarizes the experimental data on oxy-gen doped YBCO. The experimental corpus contains atotal of twenty-six distinct data sets from 1996 to 2018.Appendix C gives full details on each of these data sets,three more data sets which are in excellent agreement butwhich we have omitted because their data points showunusually large scatters , and another data set whichwe have omitted on a case by case basis.
Although oxygen doped YBCO has a considerablyhigher maximum T c ≈
94 K than LSCO, the data onits phase diagram is much curtailed. Hole dopings above p = 0 .
194 cannot be obtained at ambient pressure, andthe experimental literature on the pseudogap generallyhas not measured dopings less than p ≤ .
05, most likelybecause of proximity to the antiferromagnetic Neel phase,marked in grey in Figure 3. Whether this practice as-sumes or confirms that the pseudogap lines do not coex-ist with Neel order seems unclear. We have not collatedcharacteristic temperatures on the calcium doped vari-ant of YBCO, in which hole dopings well in excess of p = 0 .
194 can be achieved, because much less experi-mental data is available on characteristic temperaturesin this variant compound.In the remaining doping range 0 . ≤ p ≤ .
194 weare able to clearly identify the n = 2 , n = 3 , and n = 4 pseudogap lines, which follow rays intersectingnear p ≈ . The scatter of data around each char-acteristic temperature is about the same as in LSCO, oforder 15 K. We find again clear gaps between the charac-teristic temperatures, allowing each data set to be unam-biguously assigned to one of the characteristic tempera-tures.The orange lines (ten data sets) show YBCO’s pseu-dogap temperature T ∗ , marked by transport signaturesand the onset of fluctuating intra unit cell order. Re-cent symmetry-oriented experiments show that this isa nematic C symmetry breaking transition similar tothe symmetry breaking seen on LSCO’s n = 2 line.However these experiments go beyond the neutron scat-tering and X-ray diffraction experiments performed onLSCO, and find that C , mirror, and inversion sym-metries are also broken in addition to C symmetrybreaking. On thegreen T nem lines (seven data sets), transport becomes ne-matic and time reversal symmetry is broken. The nematic temperatures T nem have values equal to twothirds those of the pseudogap temperatures T ∗ , not onehalf. This is the reason why we assign n = 2 to T ∗ ratherthan n = 1, and n = 3 to T nem rather than n = 2. Atlarge dopings both the n = 2 T ∗ and n = 3 T nem linesclearly cross the superconducting dome and coexist withthe superconducting phase, excluding the possibility thatthe pseudogap transition merges with the superconduct-ing phase transition rather than crossing it. Next theblue T ∗∗ lines ( n = 4, six data sets) have been measuredin resistance and Hall resistance experiments, and theirvalues are one half those of the pseudogap temperatures T ∗ , leading us to assign n = 4 to the T ∗∗ lines. Atdopings p < .
085 intra unit cell ordering occurs on this n = 4 pseudogap line rather than on the n = 2 line. Turning to YBCO’s n = 1 pseudogap line, experimentshave almost never explored temperatures above 300 K be-cause of concerns about thermal history memory, agingand equilibration, and preparation protocol. This leavesonly a small corner in the phase diagram where the n = 1line might be seen, stretching from p = 0 . , T = 300 Kto p = 0 . , T = 95 K, which limits our ability to ascer-tain the existence of a line here. In this region there is alarge step change in the resistance and an onset of ther- T [ K ] n =1 n =2 n =3 n =4 FIG. 4: (Color online.) Intersection of the pseudogap familyin YBCO near T = 0 , p = 0 . mal history memory . Moreover Ando’s resistancemeasurements show a prominent white contour, plottedin red in Figure 3, which is roughly aligned with the ex-pected n = 1 line. All four pseudogap lines intercept T = 0 at doping p ≈ .
215 with a scatter that is again notably small, com-parable to the scatter around LSCO’s p ≈ .
26. Figure 4displays linear regressions of the twenty-two pseudogapdata sets which contain more than one data point.
The linear regressions of fifteen out of twenty-two pseu-dogap data sets intercept the T = 0 axis in the inter-val p = [0 . , . . , . p = 0 .
20, excluding the possibilitythat YBCO’s pseudogap intercept lies near p = 0 . III. COMPARISON
Comparing LSCO to YBCO, the pseudogap lines ex-trapolate to zero-doping temperatures which are thesame within experimental error, i.e. 515 K for LSCO’s n = 2 line vs. 500 K for YBCO. However the pseudo-gap family’s meeting point on the T = 0 axis lies at con-siderably different dopings: in YBCO it lies at p ≈ . p ≈ .
26. This should layto rest debates about the T = 0 intercept of the pseudo-gap line, and about the location of the putative quantumcritical point: if the question is framed in terms of holedoping p , the answer clearly depends on the material.Yet the question can be answered in a different way thatis not material dependent: in both LSCO and YBCOthe pseudogap’s T = 0 intercept lies very near to themaximal doping p c at which superconductivity can beachieved, which has two different values, p c = 0 . in LSCO vs. p c = 0 . in oxygen-doped YBCO atambient pressure. In this sense the pseudogap lines inboth LSCO and YBCO reach T = 0 at the real material-dependent end of the superconducting phase. We con-clude that in both materials the entire superconductingdome lives under the shelter of the pseudogap family, andthat its overdoped edge is tied to the pseudogap.The various observables marking the pseudogap familycollapse at critical dopings which depend on the line, andthe doping scheme , and the observable. On LSCO’s n = 1 line the linear resistance signature collapses be-tween p = 0 .
17 and p = 0 . , while the peak inthe magnetic susceptibility continues past p = 0 . .On the n = 2 line the tetragonal phase associated withthis line collapses near p = 0 . , and on the n = 3 linethe ARPES pseudogap persists until at least p = 0 . .It seems likely that the collapse of each line is causedby a small competing temperature scale, since in bothLSCO and YBCO the n = 2 , n = 3 , and n = 4 lines ei-ther collapse or disappear from the experimental recordbetween 50 and 110 K. The coupling between copper oxy-gen planes is a likely candidate for supplying the compet-ing scale, since it supports long range 3-D charge densitywaves and superconductivity in roughly the same tem-perature range.The experimental signatures associated with each ofthe pseudogap lines in YBCO could differ substantiallyfrom the signatures seen on corresponding lines in LSCO.Except for the n = 2 line, comparisons of symmetrybreaking in LSCO and YBCO are impossible becauseLSCO pseudogap experiments have generally not probedsymmetry breaking. Moreover, the hallmark of LSCO’s n = 3 line is that the ARPES density of states begins tomanifest a pseudogap, while in YBCO this change is gen-erally believed to occur instead on YBCO’s ”pseudogapline” i.e. its n = 2 line. Since there is no ARPES data onYBCO’s pseudogap temperature, it remains possible thatin YBCO the pseudogap could manifest in the density ofstates on the n = 3 line rather than the n = 2 line, thesame as in LSCO. This question is confused further bydata on p ≈ .
14 Bi Sr CaCu O δ (Bi2212) where theARPES pseudogap seems to open in two discrete stepsnear T = 250 K and T = 150 K, suggesting that inBi2212 the density of states may change substantially attwo distinct pseudogap lines. IV. ANALYSIS
These results are very fertile ground. First, the factthat the pseudogap family spans all dopings up to p c and temperatures up to 700 K argues strongly that theentire phase diagram up to p c hosts a single motherphase or order. The various phenomena observed alongthe pseudogap lines, for instance the structural phasetransition to orthorhombic order, are then subsidiary orparasitic phenomena that respond to underlying changes in the mother phase. The details of which line a par-ticular observable (such as orthorhombic symmetry) isassociated with may depend on the material, and evenwhen a particular observable in a particular materialcollapses at a particular doping the underlying line inthe mother phase may continue robustly to higher dop-ings. As a case in point, in YBCO intra unit cell or-der switches from the n = 4 line to the n = 2 linenear p = 0 . , while transport sig-nals show that the n = 4 line continues until at least p = 0 . Superconductivity may also be a sub-sidiary or parasitic phenomenon which occurs when (a)the mother phase assists hole transport and (b) the in-terlayer coupling is strong enough to support long range3-D order.Secondly, the width of the pseudogap lines is unques-tionably sharp compared to the pseudogap temperaturesthemselves. Figures 1 and 3 show that the scatter aroundeach pseudogap line is typically of order ±
15 K.
Thissharpness controverts the hypothesis that the pseudogapline is a crossover, and could be understood as evidenceof standard phase transitions. However the fact thatthere is a family of repeated pseudogap lines indicatesthat we are instead seeing a quantum coherent effect inthe mother phase, in the same category as Landau levelsor atomic orbitals. If so, then quantum coherence mustpersist to temperatures as high as 700 K in the cuprates.This conclusion is reinforced by parallel evidence fromthe strange metallic (linear in magnetic field and linearin temperature) resistance seen in the cuprates, which isalso a manifestation of quantum coherence at tempera-tures far above T c . Thirdly, the pseudogap lines are in truth linear. Clar-ity about this linearity was obtained by restricting ourdata to single materials with a single doping scheme.There are some mild deviations from linearity - YBCO’s n = 4 pseudogap line flattens at around 140 K, andLSCO’s n = 1 and n = 2 lines steepen at high dopings- but these are mild distortions, and are the exceptionrather than the rule. The pseudogap lines relate temper-ature T to the 2-D sheet density of holes ρ holes = p/ A within the copper oxide plane. Here A is the area of thecopper oxide unit cell. Since in atomic units sheet den-sity has the same units as both temperature and energy,we conclude that the pseudogap temperatures are directmeasures of a sheet density. This conclusion is supportedby the fact that in atomic units the constant of propor-tionality between the n = 1 pseudogap line and ρ holes isof order one: − .
27 for LSCO and − .
55 for YBCO. Thenatural endpoint of this reasoning is that in the cupratesthere is a two dimensional sheet density Π psg which con-trols both the pseudogap family and the family’s motherphase, that Π psg ∝ p c − p is high at low dopings anddecreases linearly to zero at p c , and that the pseudogaptemperatures are direct measures of this density.Two other linear relations between sheet density andtemperature have already been seen in the cuprates. Ue-mura’s proportionality relation between the superfluiddensity and T c holds in many underdoped materials ,and a formally identical relation has been verified also inoverdoped LSCO . Secondly, several LSCO and Bi2212experiments have reported temperature scales which riselinearly with hole doping p and extrapolate to zero at theunderdoped end of the superconducting dome, so thatthese temperatures seem to measure the sheet density ofmobile holes - see Figure 6 in Appendix A for a sum-mary of these results. It is also significant that3-D long range charge density wave (CDW) order andthe 1 / psg , sug-gesting that these may be favored when the pseudogapdensity is about one half of its maximum value.It is technically possible that the pseudogap densityΠ psg and pseudogap lines could reflect strictly ultravio-let i.e. short range physics near the atomic scale, andthat the cuprate phase diagram is not controlled directlyby Π psg but instead through renormalization group flowfrom short range to long range collective behavior. Thishas the weaknesses that atomic scale physics would beexpected to cause resonances in the phase diagram atvalues of Π psg tied to the crystal structure, but these arenot seen, and that at dopings near p c the area scale in-dicated by Π − psg is far in excess of the atomic scale. Inour view it is more likely that Π psg is a fundamental de-termining property of long-range order in the pseudogapmother phase. We leave for future work the question ofwhat quantity is counted by Π psg , although some of theobvious possibilities are vortices, skyrmions, dislocations,entanglement density, or topological quantities.Fourthly, we turn to the 1 /n quantization rule whichcontrols the slopes. In this connection we are inspired byrecent experimental and theoretical work which shows adirect linear relation between magnetic field B and tem-perature T in the strange metal phase of LSCO and otherbad metals. The pattern traced by the pseudo-gap family can be reproduced by a model of free two di-mensional p / m fermions where temperature is mappedto an effective magnetic field B and the pseudogap sheetdensity Π psg ∝ p c − p is mapped to a Fermi level E F . Asshown in Figure 5, if the effective model includes either aBerry phase or a Zeeman term µ Bσ z , then its Landaulevels have the same 1 /n slopes seen in the pseudogapfamily and intersect at a common point B = 0 , E F = 0.This model leads us to the conjecture that the pseudo-gap family is in some way related to the Integer QuantumHall Effect. Our conjecture is not a suggestion of well-defined quasiparticles or, going further, a Fermi liquid orband structure. Topology can control conduction evenwhen these concepts are not relevant, as has been seenin studies of strongly disordered topological insulators. Figure 5 shows that the effective model possesses anadditional n = 0 Landau level at E F = 0, which mapsto a vertical line in the cuprate phase diagram at theupper critical doping p c . This prediction is confirmedin LSCO by three experimental signatures which are ex-tinguished on this line: the superfluid density , the elec-tronic nematicity which is found at all lower dopings , E F ∝ Π psg ∝ p c − p B ∝ T n =0 n =1 n =2 n =3 n =4 n =5 Landau LevelsZerothFirstSecondThirdFourthFifth
FIG. 5: (Color online.) Effective model producing the samepattern seen in the pseudogap family. The model describesfree two-dimensional p / m e fermions with a Zeeman split-ting µ Bσ z moving in a magnetic field B . We plot the re-sulting Landau levels with their energies E n along the x axisand with magnetic field B along the y axis. The Landau levelslopes follow the same 1 /n pattern seen in the pseudogap fam-ily if field B is mapped to temperature and Fermi energy E F is mapped to pseudogap density Π psg . and the coefficient of the linear-in-temperature contribu-tion to the resisitivity .In summary, we have shown that the experimental cor-pus on cuprate characteristic temperatures shows thatsignals such as peaks, kinks, or extinction of diffractionpeaks are found on four pseudogap lines which span thecuprate phase diagram. In the large gaps between thefour lines no such signal is found and temperature de-pendence is smooth. This observation is also a predic-tion about future measurements of cuprate characteris-tic temperatures. We have argued that the entire cupratephase diagram from zero doping out to p c hosts a singlemother phase which is controlled by a two dimensionalsheet density, and that the observed family of pseudogaplines are subsidiary phenomena caused by changes in themother phase. We also suggest that the superconductingphase and charge density wave order are supported bythe mother phase and occur when it is augmented withan interplane coupling and, in the case of superconduc-tivity, with hole carriers. Appendix A: Temperature scales that riseproportionally to hole doping
With the exception of Refs. , the data sets graphedin Figure 6 are temperatures that rise linearly with dop-ing and extrapolate to T = 0 near the underdopedlimit of the superconducting dome, i.e. in the interval p = [0 . , . reports an energy scale with a similar be-havior.Ref. also reports a temperature which rises linearlyup to T = 300 K. This last data set extrapolates to T = 0at a lower doping p = 0 . Angularly integrated ARPES.LSCO. The authors take the first derivative of thespectrum with respect to energy, and identify apeak in the first derivative. The temperature re-ported here, which they call a coherence tempera-ture, is a break in the temperature dependence ofthe peak position. We omit a data point at p = 0 . T = 364 K. Roughly consistent with Ref. .The slope is 6200 K without the p = 0 .
15 datapoint, or 4900 K with it.2. Ino, 2009. Angularly integrated ARPES. LSCO.The quantity reported here is a measure of thewidth of the Fermi surface. Unlike all other datadiscussed in this article, this is an energy scale con-verted to temperature, not an experimental tem-perature. The experiment was performed at T = 18K. The quantity reported here includes a factor of1 /π which might be able to be renormalized atwill. We include this data set because the fourdata points between p = 0 .
074 and p = 0 .
203 riselinearly with doping and extrapolate to the un-derdoped edge of the superconducting dome. Weomit the p = 0 .
30 point from the linear fit, and weomit the p = 0 point altogether because Figure 2 inRef. shows p = 0 data that seems to leave littleground for extracting a width. The slope is 4600K.3. Kim, 2004. Thermoelectric power. LSCO. Thetemperature reported here marks a break from thelinear signal seen at high temperatures. Here weplot only the dopings at p = 0 .
20 and higher. Theslope is 2100 K, about half of the slope in Ref. .4. Chatterjee, 2011. ARPES. Bi2212. Below thistemperature the spectrum contains a sharp Gaus-sian peak, and above this temperature the peak is T [ K ] T Neel
Panagopoulos - HysteresisHashimoto, AIARPESIno, ARPESKim, TEPChatterjee, ARPESKim, TEPOhsugi, WeissT c FIG. 6: (Color online.) Characteristic temperatures thatseem to be direct manifestations of the density of mobileholes. Light green, red, and blue data sets were measuredusing ARPES, yellow and light purple sets with the thermo-electric power, and the dark purple data set with the magneticsusceptibility. The dark green set measuring the onset of mag-netic hysteresis intercepts the p axis at lower doping, perhapsbecause it is sensitive to pinned holes. The superconductingdome is shown in pink, and the antiferromagnetic phase ingrey. absent. Unlike all other data discussed in this arti-cle, this data is obtained from Bi2212.5. Kim, 2004. Thermoelectric power. LSCO. Thetemperature reported here marks a break from thelinear signal seen at low temperatures. Here weplot only the dopings at p = 0 .
20 and higher.6. Ohsugi, 1991. Nuclear quadrupole resonance.LSCO. The temperature reported here is a Weisstemperature obtained by fitting the nuclear spinrelaxation rate to a Curie-Weiss law.7. Panagopoulos, 2006. See also Ref. by the samegroup. LSCO. This temperature marks the onsetof hysteresis in the temperature dependence of thelow field magnetization, which is probably a signof pinned vortices and of pairing. The low-dopingdata from p = 0 .
03 to p = 0 .
10 nicely follows astraight line originating at T = 0 , p = 0 .
023 andextending up to room temperature. The small p =0 .
023 intercept may be caused by the observable’ssensitivity to both pinned and mobile holes. Theslope is about 3900 K, roughly comparable to theslopes of the LSCO ARPES data sets.
Appendix B: LSCO Data Sets
The temperatures gathered here were all realized ex-perimentally rather than by extrapolation from lowertemperatures, and clearly identifiable signals occurred atthe reported temperatures, including peaks, kinks, ex-tinction of diffraction peaks, etc. In the interest of claritywe do not rely on universality arguments, and thereforerestrict ourselves to lanthanum cuprate with strontiumdoping(LSCO), and we keep separate the results of dis-tinct experimental groups and of distinct experimentalprobes and signatures. We do not report any tempera-tures that are not already reported by the articles we havecited. In particular, we have stayed out of the business ofre-analyzing or fitting data sets from other articles. Theone exception to this rule is our use of the color maps in -from that article we extracted data from certain contoursand features that are prominent in the color maps.Our survey of pseudogap temperature measurementsdoes not extend to the extensive literature on anoma-lies and phase transitions measured using mechanically-oriented observables such as internal friction, sound ve-locity, lattice constants, thermal expansivity, and thelike. All data sets used in the LSCO figures or enumeratedhere, with discussion of their particulars and origin, andwith a script that produces the figures, are available inthe supporting material as a python script.1. n = 1 line: In LSCO the n = 1 line marks a res-onance in the magnetic susceptibility and qualita-tive changes in transport and in the thermoelectricpower.(a) Yoshizaki, 1990. Peak in the magnetic sus-ceptibility.(b) Nakano, 1994. Peak in the magnetic suscep-tibility. The peak continues past p = 0 .
20, anddisappears near p = 0 .
22. We omit the lastthree data points because no peak exists inthose susceptibility curves until a Curie termhas been subtracted. We also omit the firstfour data points, at lowest doping, becausethe experimental temperature did not go highenough to see the actual peak. The scalinganalysis used to obtain those four points ishowever very convincing, especially becausethe peak was seen at lower dopings by .(c) Kim, 2004. Thermoelectric power. The tem-perature reported here marks a break from thelinear signal seen at high temperatures. Herewe plot only the dopings up to p = 0 . Resistivity. The temperaturereported here marks a break from the linearsignal seen at high temperatures. The breakdisappears and the signal seems to be perfectlylinear (from visual inspection) at p = 0 .
16 and p = 0 .
18, which is consistent with Ref. ’s data at p = 0 . , .
18. This is also consis-tent with Ref. ’s statement that the resistiv-ity signal of the pseudogap collapses between p = 0 .
17 and p = 0 . and Ref. , which was cited by Ref. by the same authors. Ref. reports that at p = 0 .
14 and p = 0 .
16 the pseudogap temper-ature, i.e. the onset of linear in temperatureresistance, occurs at 471 K and 317 K respec-tively. In contrast Ref. gives resistivity dataat p = 0 . , . , . , .
163 that super-ficially indicates much smaller temperatures.The p = 0 .
136 resistivity seems to be linearabove 150 K or so, and the p = 0 .
143 resistiv-ity seems to be linear above 110 K. HoweverRef. ’s data goes up to only 200 K, so theyare unable to detect any linearity above 471K. In contrast, Ref. ’s data at p = 0 .
14 ex-tends up to 900 K. Moreover, Ref. ’s dataextends down to the superconducting temper-ature, shows that the slope below 471 K is notvery different from the slope above that tem-perature, and changes only gradually. There-fore Ref. may just not have enough data todetect nonlinearity near the high end of theirtemperature range. A second alternative isthat there could be two linear regimes, oneabove 471 K, and another above 110 - 140K, corresponding to two pseudogap temper-atures.2. n = 2 line: In LSCO the n = 2 line is a symmetrybreaking transition from C down to C nematic or-der. It has been commonly regarded as a structuraltransition from tetragonal to orthorhombic symme-try. It is accompanied by features in transport andin the thermoelectric power.(a) Kim, 2004. Thermoelectric power. The tem-perature reported here marks a break from thelinear signal seen at low temperatures.(b) Takagi, 1992. X-ray diffraction, looking fora peak splitting caused by orthorhombicity.This data shows that the signal i.e. the or-thorhombic phase collapses near p = 0 . We reproduce a redlinear feature that is very prominent in Ando’splot. The feature disappears in the range be-tween p = 0 .
178 and p = 0 . Neutron scattering sensitiveto orthorhombic symmetry.(e) Keimer, 1992. Neutron scattering sensitiveto orthorhombic symmetry. More specifically,extinction of the (021) superlattice reflectionpeak.3. n = 3 line: In LSCO the n = 3 line, also calledthe pseudogap temperature, is marked by the birthof a pseudogap controlling the density of states. Itis accompanied by features in transport and in thethermoelectric power.(a) Yoshida, 2012. ARPES. The data here is thetemperature where the pseudogap (measuredwith ARPES) disappears.(b) Ando, 2004. Resistivity. We reproduce a linesegment imposed on the data by the authorswhich follows qualitative trends of the colorsin their graph. We omit the first of the twoline segments which they imposed, a constant-temperature T = 298 K line at low doping p ≤ . T = 298 K line is in factsupported by colors in the graph. At theoverdoped end, the authors’ line continues to p = 0 . , T = 125 K, but the underlyingresisitivity data suggests that the line shouldstop a little earlier, between p = 0 .
15 and p = 0 . Peak in the thermoelectricpower. Here we plot only the dopings from p = 0 .
05 up to p = 0 .
20. At higher dopingsthe signal increases very slowly with doping.(d) Hashimoto, 2007. Angularly integratedARPES. At temperatures below this temper-ature the density of states increases relativelyrapidly with temperature, and above this tem-perature the DOS increases less quickly withtemperature - i.e. the slope changes fromone value to another in a discontinuous way.We omit a zero temperature data point atp=0.30 because it only bounds the dopingvalue where this line collapses to the range be-tween p = 0 .
22 and p = 0 .
30. We also omit the p = 0 . , T = 300 K data point because hereno break is observed in the data, i.e. thereis no data above 300 K. This pseudogap tem-perature is based on the authors noticing thatat other dopings the slope below T ∗ decreasesinversely with T ∗ , as seen in the inset of theirFigure 2a. This p = 0 .
03 data point agreesvery well with all other data points on thisline, and extends the line down to p = 0 . ARPES. Based on measure-ments of the normal-state antinodal spectralgap - T ∗ is the temperature where this gapgoes to zero. This data point is omitted be-cause it concerns Nd-LSCO, but it does matchthe n = 3 line well. They also show a strongpseudogap at p = 0 .
12 which results in abound T ∗ >
75 K, though probably T ∗ ismuch larger than 75 K.4. n = 4 line: In LSCO the n = 4 line is markedby features in transport and in nuclear magnetic resonance.We plot only the first three data sets listed here.The same pseudogap line revealed by these threedata sets is very well supported by resistivity andNernst data from Nd-LSCO and Eu-LSCO, as doc-umented in the remaining data sets. All resistivityand Nernst data sets are shown together in Ref. and show impressive agreement.(a) Cyr-Chinoire, 2018 re-analyzes Fujii,2010. Nernst effect. The tempera-ture reported here marks a break from thelinear signal seen at high temperatures.(b) Cyr-Chinoire, 2018 re-analyzes Ong,2010/2011.
Nernst effect. The temper-ature reported here marks a break from thelinear signal seen at high temperatures. Thisdata set is a re-analysis of Nernst effect datafrom two papers which have Wang, Xu, Ong,and Uchida as co-authors.(c) Itoh, 2004. NMR. Peak in the nuclear spin-lattice relaxation rate.(d) Cyr-Choiniere, 2018.
Nernst effect.The temperature reported here marks a breakfrom the linear signal seen at higher tempera-tures. This data set of three data points from p = 0 .
15 to p = 0 .
21 is omitted because itconcerns Nd-LSCO, but it matches the LSCO n = 4 line well. It includes one data pointfrom Ref. . It also includes two data pointsfrom Cyr-Choiniere’s Ref. . A third p = 0 . T = 0 and therefore it only bounds thedoping value rather than fixing it - howeverthis data point does show that in Nd-LSCOthis pseudogap line collapses somewhere be-tween p = 0 .
21 and p = 0 . Resistivity. The tempera-ture reported here marks a break from the lin-ear signal seen at higher temperatures. Thisdata set from p = 0 .
20 to p = 0 .
24 is omittedbecause it concerns Nd-LSCO, but it matchesthe n = 4 line well. When fitting to a straightline, we (doubly) omit a T = 0 data pointat p = 0 .
24. The pseudogap temperature at p = 0 .
23 is 40 K, so the pseudogap temper-ature collapses to zero between p = 0 .
23 and p = 0 . Resistivity. The temperature reported heremarks a break from the linear signal seenat higher temperatures. This data set from p = 0 .
12 to p = 0 .
15 is omitted because itconcerns Nd-LSCO, but it matches the n = 4line well. This data set is two data points ob-tained by re-analyzing data from Ichikawa.0(g) Cyr-Chinoire, 2018. Nernst effect. Thetemperature reported here marks a break fromthe linear signal seen at higher temperatures.This data set from p = 0 .
08 to p = 0 .
21 isomitted because it concerns Eu-LSCO, but itmatches the LSCO n = 4 line well. One datapoint within this data set is obtained from re-analysis of work from separate authors, so we(doubly) omit it. Three data points are fromRef. and one data point is from Ref. bythe same authors.5. Other Data Sets:(a) Baledent, 2010. Neutron scattering detectingintra unit cell two-dimensional short range or-der. This data point lies 4K above the n = 6line, well within experimental error bars.(b) Itoh, 2004. NMR. A minimum in the nu-clear spin-lattice relaxation rate. This dataset looks like it could belong to the 5th or n = 6 line.(c) Xu, 2000. Tangent of the Hall angle. Thetemperature reported here marks a break froma quadratic form which gives a good fit athigher temperatures. We omit a data pointwhich lies at T = 0 , p = 0 .
17 because itonly bounds the doping value where this tem-perature goes to zero to the interval between p = 0 .
10 and p = 0 . Peak in the heat capac-ity divided by temperature. This data set liesnear the n = 5 and n = 6 lines.6. Superconducting T c and Neel temperature:(a) Momono, 1994. T c .(b) Ando, 2004. T c .(c) Doiron-Leyraud, 2009. T c = 0 at p = p c =0 . Neel temperature measuredwith neutron scattering.(e) Matsuda, 2002. Neel temperature measuredwith neutron scattering.(f) Niedermayer, 1998. Neel temperature mea-sured with muon spin rotation.7. Omitted Data Sets:(a) Panagopoulos, 2004, 2005, 2006.
Onsetof hysteresis in the temperature dependencein the low field magnetization. We omit thesedata sets because they are clearly not relatedto the pseudogap family. They are look like ahigh-temperature replica of the superconduct-ing dome, reaching a maximum T ≈
300 K,and cut across the n = 2 line and all lowerlines. (b) Oda, 1990. Peak in the magnetic suscepti-bility. We omit this data set because it seemshave been superceded by Ref. by the sameauthors.(c) Oda, 1990. Peak in the magnetic suscepti-bility. We omit this data set because it dupli-cates data in Ref. by the same authors.(d) Oda, 1991. Peak in the magnetic susceptibil-ity. We omit this data set because it seems tohave been superceded by Ref. by the sameauthors. It does indicate that the peak disap-pears between p = 0 .
19 and p = 0 . Peak in the magnetic sus-ceptibility. We omit this data set because itseems to duplicate data in Ref. , which waspublished in the same year by the same au-thors.(f) Nakano, 1998. Peak in the magnetic suscep-tibility. We omit this data set because five outof six data points are the same as Ref. by thesame authors, but divided by 4 . Magnetic susceptibility. Thetemperature reported marks a break from thelinear form seen at high temperatures. Weomit this data set because it is the same asthe susceptibility data in Ref. by the sameauthors, divided by 4 . Resistivity. We omit thisdata set it is roughly a factor of six smallerthan the resistivity data in Ref. by the sameauthors. The authors probably divided thisdata by some number to map to an energyscale, just as they did with the magnetic sus-ceptibility data.(i) Hwang, 1994. Hall resistance. We omit thisdata set because the method used to obtainit seems to allow for renormalization of theentire data set by a somewhat arbitrary mul-tiplicative factor.(j) Batlogg, 1994. Hall resistance. We omit thisdata set because the method used to obtainit seems to allow for renormalization of theentire data set by a somewhat arbitrary mul-tiplicative factor. The authors are the same asin Ref. and the Hall resistance data looks thesame too. However here the data are limitedto p = 0 .
15 and higher while the other articleincludes five additional dopings that are lessthat p = 0 . Reanalysis of Yoshizaki’smagnetic susceptibility data, looking for ashoulder rather than a peak. We omit thisdata set because for the two highest doping1data points the shoulder is hard to identifyand because at the four lower doping datapoints the physical meaning of the shoulder isnot clear, since it remains to be seen whetherthere is actually a peak at those dopings.(l) Batlogg, 1994. Resistivity - ”changes of thehigh temperature slope curves.” We omit thisdata set because it is only at lower dopings andits mathematical and physical meaning, i.e.the precise experimental meaning of ”changesof the high temperature slope curves”, is notclear.(m) Tallon, 1999, analyzes Boebinger, 1996.
Resistivity. We omit this data set because themethod used to obtain it seems to allow forrenormalization of the entire data set by anarbitrary multiplicative factor.(n) Johnston, 1989. Peak in the magnetic sus-ceptibility. While this paper is distinguishedby being perhaps the first to notice a spinpseudogap, only three data points concernstrontium doped LSCO. Of these three points,only one is from data that actually showed apeak in the susceptibility. We omit the re-maining data point.(o) Takemura, 2000, analyzes Nishikawa,1994.
Thermoelectric power, analyzedwith a universal scaling method. We donot plot this data because the temperaturereported here lies in the tail of the universalscaling curve and is not associated with anyclear feature. On the other hand the resultshere are roughly the same as those producedby ’s thermoelectric power experiment,which used the onset of a linear signal todefine their characteristic temperature.(p) Startseva, 1999. Optical reflectivity and con-ductivity. We omit this data because there areonly two data points at dopings separated byonly 0 .
01 (probably close to the error bars indoping), and because the difference in temper-ature is very large: 50 K.(q) Wang, 2006. Nernst effect. The tempera-ture reported here marks a break from thelinear signal seen at higher temperatures. Weomit this paper and several others by the samegroup because their data has recently beenquestioned and re-interpreted by Ref. . Appendix C: YBCO Data Sets
The temperatures gathered here were all realized ex-perimentally rather than by extrapolation from lowertemperatures, and clearly identifiable signals occurred at the reported temperatures, including peaks, kinks, ex-tinction of diffraction peaks, etc. In the interest of clar-ity we do not rely on universality arguments, and there-fore restrict ourselves to oxygen doped YBCO, and wekeep separate the results of distinct experimental groupsand of distinct experimental probes and signatures. Wedo not report any temperatures that are not already re-ported by the articles we have cited. In particular, wehave stayed out of the business of re-analyzing or fittingdata sets from other articles. The one exception to thisrule is our use of the color maps in Ref. - from that arti-cle we extracted data from certain contours and featuresthat are prominent in the color maps.Our survey of pseudogap temperature measurementsdoes not extend to the extensive literature on anoma-lies and phase transitions measured using mechanically-oriented observables such as internal friction, sound ve-locity, lattice constants, thermal expansivity, and thelike. Nor did we explore the extensive literature onhysteresis in YBCO and associated onset temperatures,or on oxygen movement and ordering.When the hole doping was not reported, we used Ref. to map from oxygen content (or, in one case, from T c ) tohole doping.All data sets used in the YBCO figures or enumeratedhere, with discussion of their particulars and origin, andwith a script that produces the figures, are available inthe supporting material as a python script.1. n = 1 line:(a) Ando, 2004. Resistivity. We reproduce awhite contour that is prominent in Ando’sdata and starts near p = 0 . , T = 291 K.2. n = 2 line: In YBCO the n = 2 line is marked bybroken C , C , mirror, and inversion symmetries,by fluctuating intra unit cell order which includestime reversal symmetry breaking, by a new con-tribution to nematic ordering, and by signaturesin transport and in crystal vibrational frequencies.Several of the symmetry breaking signatures dropto the n = 4 line at dopings lower than p < . Muon spin relaxation. Attemperatures below this temperature the spinrelaxation reflects the existence of a slowlyfluctuating magnetic field consistent with theintra unit cell order seen by neutron scatter-ing, while above this temperature the field isabsent. The entirety of this data is contestedby Ref. and a reply to that is containedwithin Ref. .(b) Sato, 2017. Torque magnetometry measure-ments of the anisotropic susceptibility. Abovethis temperature the anisotropy is a linearly2increasing function of temperature, while be-low this temperature it begins increasing astemperature decreases. This signals that atthe pseudogap a new contribution to nematicorder is added.(c) Zhao, 2016.
Linear and n = 2 line opti-cal anisotropy. Below this temperature the n = 2 line appears, signaling loss of inversionand C two-fold rotation symmetries. This isa stronger symmetry breaking than either ne-matic order or orthorhombic symmetry.(d) Zhao, 2016, analyzes Lubashevsky, 2014. Optical birefringence. The temperature re-ported here marks the onset of a polarized sig-nal seen at lower temperatures, which signalsloss of both mirror and C four-fold rotationsymmetries.(e) Sidis-Bourges, 2006-2015. Polar-ized neutron scattering showing the onset ofintra unit cell order, i.e. time reversal sym-metry breaking while retaining lattice transla-tional symmetry. Thirteen points from six dif-ferent papers. The lowest doping data point,at p = 0 . , T = 170 K, lies near the n = 4line (about 13 K higher, within the publishederror bars of ±
30 K), while the other twelvepoints at higher dopings lie on the n = 2 line.We omit this because the data set shows ascatter (compared to its linear regression, andnot including the p = 0 .
08 data point) of 30K, which is above our cutoff.(f) Daou, 2018/2010, analyzes Ando, 2004.
Resistivity. The temperature reported heremarks a break from the linear signal seen athigher temperatures. We omit the p = 0 . .(g) Arpaia, 2018. Thin film on an MgO sub-strate. Resistivity. The temperature reportedhere marks a transition to linearity at highertemperatures.(h) Arpaia, 2018. Thin film on an SrTiO sub-strate. Resistivity. The temperature reportedhere marks a transition to linearity at highertemperatures.(i) Alloul, 2010. Resistivity. The temperaturemeasured here marks a transition to the linearresistivity seen at higher temperatures. Weomit the p = 0 .
169 data point because thesample has been irradiated to produce moredisorder.(j) Wang, 2017. Resistivity. The tempera-ture recorded here marks the transition toquadratic behavior at low temperatures.(k) Kabanov, 1999. Thin films on Mg0 andSrTiO . Photoinduced transmission. The temperature measured here marks the end ofa low temperature plateau in δT /T , where T is the photoinduced transmission signal. Weomit this because the data shows a scatter(compared to its linear regression) of 30 to 40K, which is above our cutoff.(l) Leridon, 2009. First derivative of the mag-netic susceptibility with respect to tempera-ture. At temperatures above this temperaturethe derivative is a decreasing function of T ,while at lower temperatures it reverses its be-havior and begins decreasing as T is reduced.We omit this data set because the data showsa scatter (compared to its linear regression) of30 to 40 K, which is above our cutoff.(m) Shekhter, 2013. Resonant ultrasound spec-troscopy measuring crystal resonance frequen-cies. At this temperature a sharp anomaly isseen in the resonance frequency and its slopechanges abruptly. The width of the anomalyis 3 K, which is far sharper than any otherdata set on the pseudogap.It is worth noting that, unlike the otherpseudogap signatures discussed in this articlewhich focus on electronic response, resonantultrasound spectroscopy is a probe of ionicmotion. As such it belongs to the extensivecuprate literature on mechanically-orientedobservables such as internal friction, soundvelocity, lattice constants, thermal expansiv-ity, oxygen movement, thermal history depen-dence, and the like.
It is extremely wellattested that these observables reveal manydistinct anomalies and phase transitions in thetemperature range between T c and room tem-perature. These features, their dependence ondoping, and their qualitative behavior are notyet well understood. In this connection Ref. has contested Ref. ’s pseudogap data in itsentirety.3. n = 3 line: In YBCO the n = 3 line is marked bytime reversal symmetry breaking, onset of a spinresonance, and a new contribution to nematic orderas seen in transport.(a) Kapitulnik, 2009. This temperature marksthe onset of the Kerr effect, signaling time re-versal symmetry breaking. We omit the datapoint at p = 0 .
156 because it has an enormous87 K error bar.(b) Xia, 2008. Polar Kerr effect. This tem-perature marks the onset of the Kerr effect,signalling time reversal symmetry breaking.They also find hysteresis at these tempera-tures, up to room temperature, indicating thattime reversal symmetry breaking occurs alsoup to room temperature. We omit this data3set because three out of four data points arerepeated in by the same authors.(c) Dai, 1999. Neutron scattering. The tem-peratures recorded here mark the onset of amagnetic resonance which is measured by in-tegrating the magnetic structure factor overmomentum and frequency.(d) Cyr-Choiniere, 2015. Nematic component ofthe Nernst effect. The temperature recordedhere marks a transition from steeply decreas-ing behavior (at low T) to slowly increasinglinear behavior (at higher T). The authors ar-gue that this marks the onset of a new contri-bution to the nematicity, and that this con-tribution is distinct from the nematicity athigher dopings which may be associated withcharge density waves.(e) Cyr - Choiniere, 2015, analyzes Ando,2004.
Nematic component of the resistiv-ity. They plot the ratio ρ a /ρ b , where ρ a and ρ b are measured along two different axis. Thetemperature recorded here marks a transitionfrom steeply decreasing behavior (at low T)to slowly increasing linear behavior (at higherT). The authors argue that this marks the on-set of a new contribution to the nematicity,and that this contribution is distinct from thenematicity at higher dopings which may beassociated with charge density waves.(f) Cyr - Choiniere, 2015. Nematic componentof the resistivity. They plot the ratio ρ a /ρ b ,where ρ a and ρ b are measured along two dif-ferent axis. The temperature recorded heremarks a transition from a steeply decreasingbehavior (at low T) to slowly increasing be-havior (at higher T). The authors argue thatthis marks the onset of a new contribution tothe nematicity, and that this contribution isdistinct from the nematicity at higher dopingswhich may be associated with charge densitywaves.(g) Wuyts, 1996. Resistivity. They take thederivative of the resistivity with respect totemperature, find a peak in the derivative, andthe peak position is the temperature recordedhere. They originally multiplied by two andwe remove that factor.4. n = 4 line: In YBCO the extent of the n = 4 line,from p = 0 .
053 to p = 0 . Thin film on an MgO sub-strate. Resistivity. The temperature reportedhere marks a transition from quadratic atlower temperatures. Data from p = 0 .
053 to p = 0 . Thin film on an SrTiO sub-strate. Resistivity. The temperature reportedhere marks a transition from quadratic atlower temperatures. Data from p = 0 .
067 to p = 0 . Hall resistance. This temperature records thepoint where the second derivative of the Hallresistance changes sign. Data from p = 0 . p = 0 . p = 0 .
052 to p = 0 .
082 the n = 4 line is augmented by intra unit cell order,spontaneous magnetic fields, and new nematic or-der. At higher dopings similar signals are seen onthe n = 2 line.(a) Haug, 2010. Neutron scattering. Thetemperature reported here marks onset ofanisotropy in a neutron scattering triple-axisexperiment. This is a sign of nematic order,and is called an electronic liquid crystal.(b) Baledent, 2011. Polarized neutron scatteringshowing the onset of intra unit cell order, i.e.time reversal symmetry breaking while retain-ing lattice translational symmetry. The low-est doping data point, at p = 0 . , T = 170K, lies near the n = 4 line (about 13 K higher,within the published error bars of ±
30 K).(c) Sonier, 2001. Muon spin relaxation. Thisis the extinction temperature of a signal thatindicates the presence of small spontaneousmagnetic fields. Two data points. The firstlies on the n = 4 line. We omit the seconddata point, which lies about 35 K below the n = 4 line, well outside the experimental errorbars of ±
10 K, and close to the three dimen-sional charge density waves .5. Other Data Sets:(a) Arpaia, 2018. Thin film on an MgO sub-strate. Resistivity. The temperature reportedhere marks a transition to quadratic at highertemperatures. We omit this data from thelinear regressions because it clearly has twoparts, one at lower doping which decreasesvery steeply until it hits the superconductingdoping, and a second part which follows thesuperconducting dome.(b) Arpaia, 2018. Thin film on an SrTiO sub-strate. Resistivity. The temperature reportedhere marks a transition to quadratic at highertemperatures.6. Superconducting T c , Neel temperature, and chargedensity waves:(a) Coneri, 2010. Neel temperature measuredwith muon spin rotation.4(b) Coneri, 2010. T c measured with muon spinrotation.(c) Liang, 2006. T c .(d) Laliberte, 2018. Sound velocity measure-ments of 3 − D charge density wave order. Weomit the T = 0 data points at either side ofthe dome.7. Omitted Data Sets:(a) Hinkov, 2008. Neutron scattering. Polarizedneutron scattering showing the onset of intraunit cell order, i.e. time reversal symmetrybreaking while retaining lattice translationalsymmetry. We omit this data point becauseit seems to have been revised from T = 150 Kto T = 170 K in Ref. by the same authors..(b) Cyr-Choiniere, 2018 and Daou, 2010. Nernst effect. The temperature reported heremarks a break from the linear signal seen athigher temperatures. In it is shown thatthis temperature is the point where anisotropyin the Nernst coefficient is extinguished, at p = 0 . , . , . , .
18. The authors arguethat this anisotropy is caused by rotationalsymmetry breaking in the copper oxide planesas opposed to the oxygen chains. Ref. by thesame authors, five years later, re-evaluates thedata, and says that the Nernst anisotropy seenin Ref. ”is more likely to be caused by CDWmodulations” instead of the pseudogap. Inother words, although they saw the beginningof a slight rise in the Nernst anisotropy at thepseudogap temperature, the real rise doesn’toccur until near the lower temperatures whereCDW order is observed using X-ray diffrac-tion. Ref. is an effort to sort out where thenew nematicity begins. We omit this data set (but not Ref. ) because it was later revisedand reinterpreted by the same authors. Wealso note that the data shows a scatter (com-pared to its linear regression) of 15 to 20 K,which is above our cutoff for the n = 2 line.This data set cuts across the n = 1 and n = 2lines and is rather flat.(c) Goto, 1996. Nuclear magnetic resonance.The temperature reported here marks a peakin the signal. This data set is very flat andcuts through the n = 2 and n = 3 lines. Itroughly coincides with X-ray scattering dataon short-range charge density wave order fromRef. .(d) Cooper, 1996. Several pseudogap tempera-ture data sets are reported in Figure 30. Thisdata is omitted because we don’t understandwhere it came from and therefore can not ver-ify whether it meets our selection criteria, andalso because it seems that the thermoelectricpower data in Figure 30 (which runs up to 600K) was derived from data in Figure 28b, whichhas a temperature cutoff of 300 K.
Acknowledgments
We gratefully acknowledge stimulating discussionswith K. S. Kim, J. Zaanen, A. Ho, S. Hayden, M. Su-langi, K. Schalm, V. Cheianov, T. Ziman, I. Vishik, J.Saunders, L. Levitin, J. Koelzer, P. Hasnip, M. Ma, P.Abbamonte, A. Kim, P. Coleman, S. Hayden, and espe-cially A. Petrovic. We thank the Instituut Lorentz inLeiden for hospitality, and the Hubbard Consortium forfacilitating discussions. We acknowledge support fromEPSRC grant EP/M011038/1. ∗ Electronic address: [email protected] H Alloul, F Rullier-Albenque, B Vignolle, D Colson, andA Forget. Superconducting fluctuations, pseudogap andphase diagram in cuprates.
EPL (Europhysics Letters) ,91(3):37005, 2010. Yoichi Ando, Seiki Komiya, Kouji Segawa, S. Ono, andY. Kurita. Electronic phase diagram of high- T c cupratesuperconductors from a mapping of the in-plane resistiv-ity curvature. Phys. Rev. Lett. , 93:267001, Dec 2004. Riccardo Arpaia, Eric Andersson, Edoardo Trabaldo,Thilo Bauch, and Floriana Lombardi. Probing the phasediagram of cuprates with yba cu o − δ thin films andnanowires. Phys. Rev. Materials , 2:024804, Feb 2018. V. Bal´edent, B. Fauqu´e, Y. Sidis, N. B. Christensen,S. Pailh`es, K. Conder, E. Pomjakushina, J. Mesot, andP. Bourges. Two-dimensional orbital-like magnetic or-der in the high-temperature la − x sr x cuo superconductor. Phys. Rev. Lett. , 105:027004, Jul 2010. V. Bal´edent, D. Haug, Y. Sidis, V. Hinkov, C. T. Lin, andP. Bourges. Evidence for competing magnetic instabilitiesin underdoped yba cu o x . Phys. Rev. B , 83:104504,Mar 2011. B Batlogg, H. Y. Hwang, H Takagi, H. L. Kao, J Kwo,and R. J. Cava. Charge dynamics in (la, sr) cuo : fromunderdoping to overdoping. Journal of Low TemperaturePhysics , 95(1-2):23–31, 1994. S. Blanco-Canosa, A. Frano, E. Schierle, J. Porras,T. Loew, M. Minola, M. Bluschke, E. Weschke, B. Keimer,and M. Le Tacon. Resonant x-ray scattering study ofcharge-density wave correlations in yba cu o x . Phys.Rev. B , 90:054513, Aug 2014. G. S. Boebinger, Yoichi Ando, A. Passner, T. Kimura,M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku,N. Ichikawa, and S. Uchida. Insulator-to-metal crossover in the normal state of la − x sr x cuo near optimum doping. Phys. Rev. Lett. , 77:5417–5420, Dec 1996. I Boˇzovi´c, X He, J Wu, and A T Bollinger. Dependenceof the critical temperature in overdoped copper oxides onsuperfluid density.
Nature , 536(7616):309, 2016. Utpal Chatterjee, Dingfei Ai, Junjing Zhao, StephanRosenkranz, Adam Kaminski, Helene Raffy, Zhizhong Li,Kazuo Kadowaki, Mohit Randeria, Michael R Norman,and J C Campuzano. Electronic phase diagram of high-temperature copper oxide superconductors.
Proceedingsof the National Academy of Sciences , 108(23):9346–9349,2011. C. Collignon, S. Badoux, S. A. A. Afshar, B. Michon,F. Lalibert´e, O. Cyr-Choini`ere, J.-S. Zhou, S. Liccia-rdello, S. Wiedmann, N. Doiron-Leyraud, and LouisTaillefer. Fermi-surface transformation across the pseu-dogap critical point of the cuprate superconductorla . − x nd . sr x cuo . Phys. Rev. B , 95:224517, Jun 2017. F. Coneri, S. Sanna, K. Zheng, J. Lord, and R. De Renzi.Magnetic states of lightly hole-doped cuprates in the cleanlimit as seen via zero-field muon spin spectroscopy.
Phys.Rev. B , 81:104507, Mar 2010. J R Cooper and J W Loram. Some correlations be-tween the thermodynamic and transport properties ofhigh t c oxides in the normal state. Journal de PhysiqueI , 6(12):2237–2263, 1996. JR Cooper, JW Loram, Ivan Kokanovi´c, JG Storey, andJL Tallon. Pseudogap in yba 2 cu 3 o 6+ δ is not boundedby a line of phase transitions: thermodynamic evidence. Physical Review B , 89(20):201104, 2014. R A Cooper, Y Wang, B Vignolle, O J Lipscombe, S MHayden, Y Tanabe, T Adachi, Y Koike, M Nohara, H Tak-agi, C Proust, and N E Hussey. Anomalous critical-ity in the electrical resistivity of la − x sr x cuo . Science ,323(5914):603–607, 2009. O. Cyr-Choini`ere, R. Daou, F. Lalibert´e, C. Collignon,S. Badoux, D. LeBoeuf, J. Chang, B. J. Ramshaw, D. A.Bonn, W. N. Hardy, R. Liang, J.-Q. Yan, J.-G. Cheng,J.-S. Zhou, J. B. Goodenough, S. Pyon, T. Takayama,H. Takagi, N. Doiron-Leyraud, and Louis Taillefer. Pseu-dogap temperature T ∗ of cuprate superconductors fromthe nernst effect. Phys. Rev. B , 97:064502, Feb 2018. O. Cyr-Choini`ere, G. Grissonnanche, S. Badoux, J. Day,D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud,and Louis Taillefer. Two types of nematicity in the phasediagram of the cuprate superconductor yba cu o y . Phys.Rev. B , 92:224502, Dec 2015. Olivier Cyr-Choiniere, R Daou, Francis Lalibert´e, DavidLeBoeuf, Nicolas Doiron-Leyraud, J Chang, J-Q Yan, J-GCheng, J-S Zhou, J B Goodenough, S Pyon, T Takayama,H Takagi, Y Tanaka, and L Taillefer. Enhancement of thenernst effect by stripe order in a high-t c superconductor.
Nature , 458(7239):743, 2009. Pengcheng Dai, Herbert A Mook, Stephen M Hayden,Gabriel Aeppli, Toby G Perring, Rodney Dale Hunt,and F Do˘gan. The magnetic excitation spectrum andthermodynamics of high-tc superconductors.
Science ,284(5418):1344–1347, 1999. R Daou, J Chang, David LeBoeuf, Olivier Cyr-Choiniere,Francis Lalibert´e, Nicolas Doiron-Leyraud, B J Ramshaw,Ruixing Liang, D A Bonn, W N Hardy, and L Taillefer.Broken rotational symmetry in the pseudogap phase of ahigh-t c superconductor. Nature , 463(7280):519, 2010. Nicolas Doiron-Leyraud, P Auban-Senzier, S Ren´e de Cotret, A Sedeki, C Bourbonnais, D Jerome, K Bech-gaard, and Louis Taillefer. Correlation between linear re-sistivity and tc in organic and pnictide superconductors. arXiv preprint arXiv:0905.0964 , 2009. J Dominec. Ultrasonic and related experiments in high- t c superconductors. Superconductor Science and Technology ,6(3):153, 1993. B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C. T. Lin,X. Chaud, and P. Bourges. Magnetic order in the pseudo-gap phase of high- T c superconductors. Phys. Rev. Lett. ,96:197001, May 2006. Eduardo Fradkin, Steven A. Kivelson, and John M. Tran-quada. Colloquium: Theory of intertwined orders in hightemperature superconductors.
Rev. Mod. Phys. , 87:457–482, May 2015. T. Fujii, T. Matsushima, T. Maruoka, and A. Asamitsu.Effect of stripe order strength for the nernst effect inla − x sr x cuo single crystals. Physica C: Superconductivityand its Applications , 470:S21 – S22, 2010. Proceedings ofthe 9th International Conference on Materials and Mech-anisms of Superconductivity. Paula Giraldo-Gallo, JA Galvis, Z Stegen, KA Modic,FF Balakirev, JB Betts, X Lian, C Moir, SC Riggs, J Wu,A T Bollinger, X He, I Bozovic, B J Ramshaw, R D Mc-Donald, G S Boebinger, and A Shekhter. Scale-invariantmagnetoresistance in a cuprate superconductor.
Science ,361(6401):479–481, 2018. Atsushi Goto, Hiroshi Yasuoka, and Yutaka Ueda. Car-rier concentration dependence of the spin pseudo-gap be-haviors in yba2cu3oy.
Journal of the Physical Society ofJapan , 65(9):3043–3048, 1996. M. Hashimoto, T. Yoshida, K. Tanaka, A. Fujimori,M. Okusawa, S. Wakimoto, K. Yamada, T. Kakeshita,H. Eisaki, and S. Uchida. Distinct doping dependences ofthe pseudogap and superconducting gap of la − x sr x Cuo cuprate superconductors. Phys. Rev. B , 75:140503, Apr2007. M. Hashimoto, T. Yoshida, K. Tanaka, A. Fujimori,M. Okusawa, S. Wakimoto, K. Yamada, T. Kakeshita,H. Eisaki, and S. Uchida. Crossover from coherentquasiparticles to incoherent hole carriers in underdopedcuprates.
Phys. Rev. B , 79:140502, Apr 2009. Daniel Haug, Vladimir Hinkov, Yvan Sidis, PhilippeBourges, Niels Bech Christensen, Alexandre Ivanov,Thomas Keller, CT Lin, and B Keimer. Neutron scatter-ing study of the magnetic phase diagram of underdopedyba cu o x . New Journal of Physics , 12(10):105006,2010. Ian M Hayes, Ross D McDonald, Nicholas P Brez-nay, Toni Helm, Philip J W Moll, Mark Wartenbe,Arkady Shekhter, and James G Analytis. Scaling betweenmagnetic field and temperature in the high-temperaturesuperconductor bafe (as − x p x ) . Nature Physics ,12(10):916, 2016. V Hinkov, D Haug, B Fauqu´e, P Bourges, Y Sidis,A Ivanov, Christian Bernhard, CT Lin, and B Keimer.Electronic liquid crystal state in the high-temperature su-perconductor yba2cu3o6. 45.
Science , 319(5863):597–600,2008. NE Hussey, RA Cooper, Xiaofeng Xu, Y Wang,I Mouzopoulou, B Vignolle, and C Proust. Dichotomyin the t-linear resistivity in hole-doped cuprates.
Philo-sophical Transactions of the Royal Society of LondonA: Mathematical, Physical and Engineering Sciences , H. Y. Hwang, B. Batlogg, H. Takagi, H. L. Kao, J. Kwo,R. J. Cava, J. J. Krajewski, and W. F. Peck. Scalingof the temperature dependent hall effect in la − x sr x cuo . Phys. Rev. Lett. , 72:2636–2639, Apr 1994. N. Ichikawa, S. Uchida, J. M. Tranquada, T. Niem¨oller,P. M. Gehring, S.-H. Lee, and J. R. Schneider. Localmagnetic order vs superconductivity in a layered cuprate.
Phys. Rev. Lett. , 85:1738–1741, Aug 2000. A. Ino, T. Mizokawa, K. Kobayashi, A. Fujimori,T. Sasagawa, T. Kimura, K. Kishio, K. Tamasaku,H. Eisaki, and S. Uchida. Doping dependent density ofstates and pseudogap behavior in la − x sr x cuo . Phys.Rev. Lett. , 81:2124–2127, Sep 1998. Y. Itoh, T. Machi, N. Koshizuka, M. Murakami, H. Ya-magata, and M. Matsumura. Pseudo-spin-gap and slowspin fluctuation in la − x sr x cuo ( x = 0 .
13 and 0.18) via Cu and
La nuclear quadrupole resonance.
Phys. Rev.B , 69:184503, May 2004. David C. Johnston. Magnetic susceptibility scaling inla − x sr x Cuo − y . Phys. Rev. Lett. , 62:957–960, Feb 1989. V. V. Kabanov, J. Demsar, B. Podobnik, and D. Mi-hailovic. Quasiparticle relaxation dynamics in supercon-ductors with different gap structures: Theory and exper-iments on yba cu o − δ . Phys. Rev. B , 59:1497–1506, Jan1999. Aharon Kapitulnik, Jing Xia, Elizabeth Schemm, andAlexander Palevski. Polar kerr effect as probe for time-reversal symmetry breaking in unconventional supercon-ductors.
New Journal of Physics , 11(5):055060, 2009. B. Keimer, A. Aharony, A. Auerbach, R. J. Birgeneau,A. Cassanho, Y. Endoh, R. W. Erwin, M. A. Kastner, andG. Shirane. N´eel transition and sublattice magnetizationof pure and doped la cuo . Phys. Rev. B , 45:7430–7435,Apr 1992. B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y.Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh,R. W. Erwin, and G. Shirane. Magnetic excitations inpure, lightly doped, and weakly metallic la cuo . Phys.Rev. B , 46:14034–14053, Dec 1992. B Keimer, S A Kivelson, M R Norman, S Uchida, andJ Zaanen. From quantum matter to high-temperature su-perconductivity in copper oxides.
Nature , 518(7538):179,2015. J. S. Kim, B. H. Kim, D. C. Kim, and Y. W. Park.Two pseudogap behavior in la − x sr x cuo : Thermoelectricpower at high temperature. Journal of superconductivity ,17(1):151–157, 2004. Rohit Kumar, Surjeet Singh, and Sunil Nair. Hightemperature linear magnetoresistance and scaling be-havior in the ba(fe − x co x ) as series. arXiv preprintarXiv:1801.03768 , 2018. F Lalibert´e, W Tabis, S Badoux, B Vignolle, D Destraz,N Momono, T Kurosawa, K Yamada, H Takagi, N Doiron-Leyraud, C Proust, and L Taillefer. Origin of the metal-to-insulator crossover in cuprate superconductors. arXivpreprint arXiv:1606.04491 , 2016. Francis Lalibert´e, Mehdi Frachet, Siham Benhabib, Ben-jamin Borgnic, Toshinao Loew, Juan Porras, Math-ieu Tacon, Bernhard Keimer, Steffen Wiedmann, CyrilProust, and David LeBouef. High field charge order acrossthe phase diagram of yba cu o y . npj Quantum Materi-als , 3(1):11, 2018. A N Lavrov. Low temperature order-disorder phenomena in yba cu o − x : an electrical resistivity study. PhysicsLetters A , 168(1):71–74, 1992. David LeBoeuf, Nicolas Doiron-Leyraud, B. Vignolle,Mike Sutherland, B. J. Ramshaw, J. Levallois, R. Daou,Francis Lalibert´e, Olivier Cyr-Choini`ere, Johan Chang,Y. J. Jo, L. Balicas, Ruixing Liang, D. A. Bonn, W. N.Hardy, Cyril Proust, and Louis Taillefer. Lifshitz criticalpoint in the cuprate superconductor yba cu o y from high-field hall effect measurements. Phys. Rev. B , 83:054506,Feb 2011. B Leridon, P Monod, D Colson, and A Forget. Thermo-dynamic signature of a phase transition in the pseudogapphase of yba cu o x high-t c superconductor. EPL (Eu-rophysics Letters) , 87(1):17011, 2009. Ruixing Liang, D. A. Bonn, and W. N. Hardy. Evaluationof cuo plane hole doping in yba cu o x single crystals. Phys. Rev. B , 73:180505, May 2006. Y. Lubashevsky, LiDong Pan, T. Kirzhner, G. Koren, andN. P. Armitage. Optical birefringence and dichroism ofcuprate superconductors in the thz regime.
Phys. Rev.Lett. , 112:147001, Apr 2014. M. Majoros, C. Panagopoulos, T. Nishizaki, andH. Iwasaki. Thermomagnetic hysteretic propertiesresembling superconductivity in the normal state ofla . sr . Cuo . Phys. Rev. B , 72:024528, Jul 2005. L Mangin-Thro, Y Sidis, A Wildes, and P Bourges. Intra-unit-cell magnetic correlations near optimal doping inyba cu o . Nature communications , 6:7705, 2015. Lucile Mangin-Thro, Yuan Li, Yvan Sidis, and PhilippeBourges. a − b anisotropy of the intra-unit-cell magneticorder in yba cu o . . Phys. Rev. Lett. , 118:097003, Mar2017. M. Matsuda, M. Fujita, K. Yamada, R. J. Birgeneau,Y. Endoh, and G. Shirane. Electronic phase separationin lightly doped la − x sr x Cuo . Phys. Rev. B , 65:134515,Mar 2002. Toshiaki Matsuzaki, Naoki Momono, Migaku Oda, andMasayuki Ido. Electronic specific heat of la − x sr x cuo :Pseudogap formation and reduction of the superconduct-ing condensation energy. Journal of the Physical Societyof Japan , 73(8):2232–2238, 2004. C. E. Matt, C. G. Fatuzzo, Y. Sassa, M. M˚ansson, S. Fa-tale, V. Bitetta, X. Shi, S. Pailh`es, M. H. Berntsen,T. Kurosawa, M. Oda, N. Momono, O. J. Lipscombe,S. M. Hayden, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough,S. Pyon, T. Takayama, H. Takagi, L. Patthey, A. Ben-dounan, E. Razzoli, M. Shi, N. C. Plumb, M. Radovic,M. Grioni, J. Mesot, O. Tjernberg, and J. Chang. Elec-tron scattering, charge order, and pseudogap physicsin la . − x nd . sr x cuo : An angle-resolved photoemissionspectroscopy study. Phys. Rev. B , 92:134524, Oct 2015. G. P. Mikitik and Yu. V. Sharlai. Manifestation of berry’sphase in metal physics.
Phys. Rev. Lett. , 82:2147–2150,Mar 1999. N Momono and M Ido. Evidence for nodes in the super-conducting gap of la − x sr x cuo . t dependence of elec-tronic specific heat and impurity effects. Physica C: Su-perconductivity , 264(3-4):311–318, 1996. N Momono, M Ido, T Nakano, M Oda, Y Okajima,and K Yamaya. Low-temperature electronic specificheat of la − x sr x cuo and la − x sr x cu − y zn y o . evidence forad wave superconductor. Physica C: Superconductivity ,233(3-4):395–401, 1994. H. A. Mook, Y. Sidis, B. Fauqu´e, V. Bal´edent, and P. Bourges. Observation of magnetic order in a supercon-ducting yba cu o . single crystal using polarized neutronscattering. Phys. Rev. B , 78:020506, Jul 2008. Peter Nagel, Volker Pasler, Christoph Meingast, Alexan-dre I. Rykov, and Setsuko Tajima. Anomalously largeoxygen-ordering contribution to the thermal expansion ofuntwinned yba cu O . single crystals: A glasslike tran-sition near room temperature. Phys. Rev. Lett. , 85:2376–2379, Sep 2000. T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura,and M. Ido. Magnetic properties and electronic con-duction of superconducting la − x sr x cuo . Phys. Rev. B ,49:16000–16008, Jun 1994. Tohru Nakano, Naoki Momono, Migaku Oda, andMasayuki Ido. Correlation between the doping depen-dences of superconducting gap magnitude 2 δ and pseu-dogap temperature t in high- t c cuprates. Journal of thePhysical Society of Japan , 67(8):2622–2625, 1998. Ch. Niedermayer, C. Bernhard, T. Blasius, A. Gol-nik, A. Moodenbaugh, and J. I. Budnick. Commonphase diagram for antiferromagnetism in la − x sr x cuo and Y − x ca x ba cu O as seen by muon spin rotation. Phys. Rev. Lett. , 80:3843–3846, Apr 1998. Takashi Nishikawa, Jun Takeda, and Masatoshi Sato.Transport anomalies of high-t c oxides above room tem-perature. Journal of the Physical Society of Japan ,63(4):1441–1448, 1994. M R Norman, D Pines, and C Kallin. The pseudogap:friend or foe of high t c ? Advances in Physics , 54(8):715–733, 2005. M Oda, H Matsuki, and M Ido. Common features ofmagnetic and superconducting properties in y-doped bi (sr, ca) cu o and ba (sr)-doped la cuo . Solid StateCommunications , 74(12):1321–1326, 1990. M Oda, T Nakano, Y Kamada, and M Ido. Elec-tronic states of doped holes and magnetic properties inla − x m x cuo (m= sr, ba). Physica C: Superconductivity ,183(4-6):234–240, 1991. M Oda, T Ohguro, H Matsuki, N Yamada, and M Ido.Magnetism and superconductivity in doped la cuo . Physical Review B , 41(4):2605, 1990. Shigeki Ohsugi, Yoshio Kitaoka, Kenji Ishida, and Ku-nisuke Asayama. Cu nqr study of the spin dynamicsin high-tc superconductor la2- xsrxcuo4.
Journal of thePhysical Society of Japan , 60(7):2351–2360, 1991. C. Panagopoulos, M. Majoros, T. Nishizaki, andH. Iwasaki. Weak magnetic order in the normal stateof the high- T c superconductor la − x sr x cuo . Phys. Rev.Lett. , 96:047002, Feb 2006. C. Panagopoulos, M. Majoros, and A. P. Petrovi´c.Thermal hysteresis in the normal-state magnetization ofla − x sr x cuo . Phys. Rev. B , 69:144508, Apr 2004. Vincent Sacksteder. Fermion loops, linear magnetoresis-tance, linear in temperature resistance, and superconduc-tivity. arXiv preprint arXiv:1801.02663 , 2018. Y Sato, S Kasahara, H Murayama, Y Kasahara, E-G Moon, T Nishizaki, T Loew, J Porras, B Keimer,T Shibauchi, and T Matsuda. Thermodynamic evidencefor a nematic phase transition at the onset of the pseudo-gap in yba cu o y . Nature Physics , 13(11):1074, 2017. Kouji Segawa and Yoichi Ando. Intrinsic hall responseof the cuo planes in a chain-plane composite system ofyba cu o y . Phys. Rev. B , 69:104521, Mar 2004. Arkady Shekhter, B J Ramshaw, Ruixing Liang, W N Hardy, D A Bonn, Fedor F Balakirev, Ross D McDonald,Jon B Betts, Scott C Riggs, and Albert Migliori. Bound-ing the pseudogap with a line of phase transitions in yba cu o δ . Nature , 498(7452):75, 2013. Yvan Sidis, Benoit Fauqu´e, Vivek Aji, and PhilippeBourges. Search for the existence of circulating currents inhigh-tc superconductors using the polarized neutron scat-tering technique.
Physica B: Condensed Matter , 397(1-2):1–6, 2007. J E Sonier, J H Brewer, R F Kiefl, R I Miller, G D Morris,C E Stronach, J S Gardner, S R Dunsiger, D A Bonn,W N Hardy, R Liang, and R H Heffner. Anomalous weakmagnetism in superconducting yba cu o x . Science ,292(5522):1692–1695, 2001. Jeff E Sonier. Comment on” discovery of slow mag-netic fluctuations and critical slowing down in the pseu-dogap phase of yba 2 cu 3 o y ”. arXiv preprintarXiv:1706.03023 , 2017. T. Startseva, T. Timusk, A. V. Puchkov, D. N. Basov,H. A. Mook, M. Okuya, T. Kimura, and K. Kishio.Temperature evolution of the pseudogap state in the in-frared response of underdoped la − x sr x cuo . Phys. Rev.B , 59:7184–7190, Mar 1999. H. Takagi, R. J. Cava, M. Marezio, B. Batlogg, J. J. Kra-jewski, W. F. Peck, P. Bordet, and D. E. Cox. Disappear-ance of superconductivity in overdoped la − x sr x cuo ata structural phase boundary. Phys. Rev. Lett. , 68:3777–3780, Jun 1992. T Takemura, T Kitajima, T Sugaya, and I Terasaki. Scal-ing behaviour of the in-plane thermopower in bi sr rcu o (r= ca, y, pr, dy and er). Journal of Physics: CondensedMatter , 12(28):6199, 2000. J L Tallon, J W Loram, G V M Williams, J R Cooper,I R Fisher, J D Johnson, M P Staines, and C Bernhard.Critical doping in overdoped high-tc superconductors: aquantum critical point?
Physica Status Solidi B , 215:531–540, 1999. Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer,J. F. Carolan, W. N. Hardy, R. Kadono, J. R. Kempton,R. F. Kiefl, S. R. Kreitzman, P. Mulhern, T. M. Rise-man, D. Ll. Williams, B. X. Yang, S. Uchida, H. Takagi,J. Gopalakrishnan, A. W. Sleight, M. A. Subramanian,C. L. Chien, M. Z. Cieplak, Gang Xiao, V. Y. Lee, B. W.Statt, C. E. Stronach, W. J. Kossler, and X. H. Yu. Uni-versal correlations between T c and n s m ∗ (carrier densityover effective mass) in high- T c cuprate superconductors. Phys. Rev. Lett. , 62:2317–2320, May 1989. I M Vishik. Photoemission perspective on pseudogap, su-perconducting fluctuations, and charge order in cuprates:a review of recent progress.
Reports on Progress inPhysics , 81(6):062501, 2018. Yayu Wang, Lu Li, and N. P. Ong. Nernst effect in high- T c superconductors. Phys. Rev. B , 73:024510, Jan 2006. Yayu Wang, Z. A. Xu, T. Kakeshita, S. Uchida,S. Ono, Yoichi Ando, and N. P. Ong. Onset of thevortexlike nernst signal above T c in la − x sr x cuo andbi sr − y la y cuo . Phys. Rev. B , 64:224519, Nov 2001. Yue Wang and Jun Zhang. Revisiting the electronic phasediagram of yba cu o y via temperature derivative of in-plane resistivity. arXiv preprint arXiv:1711.07402 , 2017. J Wu, A T Bollinger, X He, and I Boˇzovi´c. Spontaneousbreaking of rotational symmetry in copper oxide super-conductors.
Nature , 547(7664):432, 2017. B. Wuyts, V. V. Moshchalkov, and Y. Bruynseraede. Resistivity and hall effect of metallic oxygen-deficientYba cu o x films in the normal state. Phys. Rev. B ,53:9418–9432, Apr 1996. Jing Xia, Elizabeth Schemm, G. Deutscher, S. A. Kivel-son, D. A. Bonn, W. N. Hardy, R. Liang, W. Siemons,G. Koster, M. M. Fejer, and A. Kapitulnik. Polar kerr-effect measurements of the high-temperature yba cu o x superconductor: Evidence for broken symmetry near thepseudogap temperature. Phys. Rev. Lett. , 100:127002,Mar 2008. Dongwei Xu, Junjie Qi, Jie Liu, Vincent Sacksteder, X. C.Xie, and Hua Jiang. Phase structure of the topologicalanderson insulator.
Phys. Rev. B , 85:195140, May 2012. Z A Xu, N P Ong, T Kakeshita, H Eisaki, and S Uchida.Charge transport in la − x sr x cuo in the low-density limit(0.03¡ x¡ 0.10). Physica C: Superconductivity , 341:1711–1714, 2000. Z A Xu, N P Ong, Yayu Wang, T Kakeshita, andS Uchida. Vortex-like excitations and the onset of super-conducting phase fluctuation in underdoped la − x sr x cuo . Nature , 406(6795):486, 2000. K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Waki-moto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shi-rane, R. J. Birgeneau, M. Greven, M. A. Kastner, andY. J. Kim. Doping dependence of the spatially modulateddynamical spin correlations and the superconducting-transition temperature in la − x sr x cuo . Phys. Rev. B ,57:6165–6172, Mar 1998. T. Yoshida, M. Hashimoto, S. Ideta, A. Fujimori,K. Tanaka, N. Mannella, Z. Hussain, Z.-X. Shen, M. Kub-ota, K. Ono, Seiki Komiya, Yoichi Ando, H. Eisaki, andS. Uchida. Universal versus material-dependent two-gapbehaviors of the high- T c cuprate superconductors: Angle-resolved photoemission study of la − x sr x cuo . Phys. Rev.Lett. , 103:037004, Jul 2009. R Yoshizaki, N Ishikawa, H Sawada, E Kita, and A Tasaki.Magnetic susceptibility of normal state and supercon-ductivity of la − x sr x cuo . Physica C: Superconductivity ,166(5-6):417–422, 1990.
Jian Zhang, Zhaofeng Ding, Cheng Tan, Kevin Huang,Oscar O Bernal, Pei-Chun Ho, Gerald D Morris, Adrian DHillier, Pabitra K Biswas, Stephen P Cottrell, Hui Xiang,Xin Yao, Douglas E MacLaughlin, and Lei Shu. Discoveryof slow magnetic fluctuations and critical slowing downin the pseudogap phase of yba cu o y . Science advances ,4(1):eaao5235, 2018.
L Zhao, C A Belvin, R Liang, D A Bonn, W N Hardy, N PArmitage, and D Hsieh. A global inversion-symmetry-broken phase inside the pseudogap region ofyba cu o y . Nature Physics , 13(3):250, 2017.
These sets are labeled in the appendix as n = 3 set (e)and n = 4 sets (d,e,f,g). These sets are labeled in the appendix as Omitted DataSets (a,p,q).
These sets are labeled in the appendix as n = 4 sets(d,e,f,g). We place the intersection at p = 0 . , T = 0 K and alignthe second pseudogap line to the experimental data pointin at p = 0 , T = 515 K. The average value of n × dT /dp , which is approximatelyconstant because the pseudogap slope varies as dT /dp ∝ /n , is 4000 K, and the standard deviation of this valueacross the fifteen data sets in the first four pseudogap linesis 800 K. These sets are labeled in the appendix as n = 2 sets (e,k,l),and show scatters of ±
30 K, which is substantially largerthan the ±
15 K shown by the other n = 2 data sets. This is labeled in the appendix as Omitted Data Sets (c).
We align the n = 2 pseudogap line to p = 0 , T = 500 Kand p = 0 . , T = 0. We have not included the regression of one of the low tem-perature (purple) data sets which clearly does not followa linear trajectory, and instead seems to be composed oftwo line segments.
It may be worth noting that at p = 0 the n = 1 line’stemperature T = 1000 K, if converted to area A via (cid:126) A − / m e = k B T , gives an area A = 158 a . This isthree times the area of the copper oxide unit cell. Judging from authors’ estimates, comparison of data setsto their linear regressions, and comparison between datasets, most data sets show scatters or errors of ±
10 to ±
20 Kelvin. A notable exception is , which reports thatYBCO’s n = 2 pseudogap line has a width of 3 K. Thereare also several data sets with exceptionally large scat-ters of 50 K or more, including three out of four data setson LSCO’s n = 1 line and also the thermoelectric powerdata on LSCO’s n = 2 line. We have not includedin our figures three data sets on YBCO’s n = 2 pseudo-gap line because their scatters are larger than 15 K ,including neutron scattering data revealing intra unit cellorder.,including neutron scattering data revealing intra unit cellorder.