Particle-Hole Symmetry Breaking in the Pseudogap State
M. Hashimoto*1, 2, 3, R.-H. He*1, 2, K. Tanaka1, 2, 3, 4, J. P. Testaud1, 2, 3, W. Meevasana1, 2, R.
G. Moore1, 2, D. H. Lu1, 2, H. Yao1, Y. Yoshida5, H. Eisaki5, T. P. Devereaux1, 2, Z.
Hussain3, & Z.-X. Shen1, 2
1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025
2Geballe Laboratory for Advanced Materials, Departments of Physics and Applied Physics,
Stanford University, CA 94305
3Advanced Light Source, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA
4Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
5Nanoelectronics Research Institute, AIST, Ibaraki 305-8568, Japan
*These authors contributed equally to this work.
In conventional superconductors, a gap exists in the energy absorption spectrum
only below the transition temperature (Tc), corresponding to the energy price to
pay for breaking a Cooper pair of electrons. In high-Tc cuprate superconductors
above Tc, an energy gap called the pseudogap exists, and is controversially
attributed either to pre-formed superconducting pairs, which would exhibit
Work supported in part by US Department of Energy contract DE-AC02-76SF00515.
particle-hole symmetry, or to competing phases which would typically break it 1.
Scanning tunnelling microscopy (STM) studies suggest that the pseudogap stems
from lattice translational symmetry breaking 2-9 and is associated with a
different characteristic spectrum for adding or removing electrons (particle-hole
asymmetry) 2, 3. However, no signature of either spatial or energy symmetry
breaking of the pseudogap has previously been observed by angle-resolved
photoemission spectroscopy (ARPES) 9-18. Here we report ARPES data from
Bi2201 which reveals both particle-hole symmetry breaking and dramatic
spectral broadening indicative of spatial symmetry breaking without long range
order, upon crossing through T* into the pseudogap state. This symmetry
breaking is found in the dominant region of the momentum space for the
pseudogap, around the so-called anti-node near the Brillouin zone boundary.
Our finding supports the STM conclusion that the pseudogap state is a broken-
symmetry state that is distinct from homogeneous superconductivity.
The nature of the pseudogap can be explored by examining the dispersion of the
occupied electronic states measured by ARPES. As shown in Fig. 1m, when a particle-hole
symmetric gap opens from the normal state dispersion (red curve) due to homogeneous
superconductivity, one always expects an alignment between Fermi momentum kF and the
“back-bending” or saturation momentum (green arrows) of the dispersion in the gapped
states (weighted blue curve). Because of this strong constraint, the observation of back-
bending or dispersion saturation anomaly away from kF in a gapped state can be a
conclusive evidence of a particle-hole symmetry broken nature of the gap, even though the
information of the unoccupied state may be missing. Here we note that the dispersion of the
spectral peak position, regardless of the spectral weight, is the simple and direct way to
address the issue of particle-hole symmetry. Due to suppressed weight that makes it hard to
discern from experimental background, the back-bending may sometimes be subtle and
may show up as a dispersion saturation. Nethertheless, it can be distinguished from a
smooth dispersion where no dispersion saturation anomaly occurs.
Fig. 1 shows the temperature evolution across T* of the Fermi-Dirac function (FD)
divided ARPES spectra of a nearly optimally-doped cuprate superconductor
Pb0.55Bi1.5Sr1.6La0.4CuO6+δ (Pb-Bi2201) with Tc ~ 34 K. Data were taken in an antinodal cut
approximately along (π, -π)-(π, 0)-(π, π) as shown in the inset of Fig. 1g. The spectra in the
true normal state above T* (~125 K) present a parabolic dispersion of the intensity
maximum as a function of momentum with two clear Fermi level crossings at kF’s (red
spectra) and a bottom reaching EBot ~ -20 meV at (π, 0) (Fig. 1a,1g & 1n). Given that the
data were taken at 160K, the measured spectra in the true normal state are remarkably
simple, similar to that of ordinary metal 19. Whereas this is naturally expected for a band in
the absence of a gap, with the pseudogap opening around EF below T*, the spectra become
surprisingly incoherent and the spectral weight centroid is transferred towards higher
binding energy (Fig. 1a-l). Despite the broadness of the spectra at low temperatures, the
intensity maximum of each spectrum can be easily defined and traced as a function of
momentum. The dispersion thus extracted becomes stronger towards lower temperatures
with the band bottom at (π, 0) being pushed far away from the true normal state EBot as
summarized in Fig. 1n. Well below T* where the spectra are fully gapped in the measured
antinodal cut, no dispersion saturation or back-bending is observed at kF, defined by the
Fermi crossing at T = 160 K (guides to the eyes in red). Instead, while approaching EF, the
dispersion flattens and appears to bend back at momenta (green arrows) markedly away
from kF. This misalignment can be also seen in the summary of data in Fig. 1i-1l (the red
spectra are for kF and the green spectra are for possible back-bending momenta).
Contrasting to what is expected in Fig. 1m, the behaviour below T* is completely different
from the expected dispersion in homogeneous superconducting state, suggesting that the
transition from the true normal state above T* to the pseudogap state has a different origin.
In addition to the misalignment between kF and the back-bending momenta for T << T*, the
observed widening of the separation between EBot and energy at the bend-back momenta
upon lowering temperature also contradicts the expectation in Fig. 1m where the energy
separation (or the overall dispersion decreases with an opening of a superconducting gap.
The connection between the pseudogap formation and the dispersion effects may be
directly examined from the temperature dependence of spectra seen at two representative
momenta, (π, 0) and kF (Fig. 2a and 2b). With increasing temperature, the energy position
of the spectral intensity maxima continuously evolve towards and ultimately reach EBot and
EF, respectively, at T* ~ 125±10 K (Fig. 2c) where the pseudogap closes, consistent with
temperature from other reports of different measurements on Bi2201 near optimal doping 9,
10, 20, 21. The smooth temperature evolution upon cooling down from the true normal state
suggests that the dispersion of the intensity maximum found at T << T* (Fig. 1n) is directly
related to the pseudogap physics. That is, the misalignment between kF and the momenta
where back-bending of the dispersion occur at T << T* is due to the pseudogap opening.
This suggests that the pseudogap state violates the momentum structure expected from a
particle-hole symmetric homogeneous superconducting state. This finding goes beyond the
earlier STM work that suggests the lattice symmetric breaking nature of the low
temperature phase, but lacks the detailed temperature dependence linking the symmetry
breaking and pseudogap opening.
The link of the observed spectral change and pseudogap formation can also be found
in the temperature dependence of the spectral intensity itself. In Fig. 2d we plot spectral
intensity at kF in the energy window of [-0.36, -0.30] eV as a function of temperature, with
the spectra normalized either by the photon flux, which reflect raw spectral weight, or by a
selected energy window as indicated in the figure (See Supplementary Fig. 3 for the spectra
with these normalizations). Regardless of the normalization used, once again, T* is the
temperature below which a strong temperature dependence emerges. The data further
suggest that a very high energy scale is involved in the spectral weight redistribution
associated with pseudogap formation. In addition, as energy-integrated spectral weight is
conserved at a fixed momentum 14, a change in spectral intensity over a wide occupied
energy window at kF could involve spectral weight redistribution from unoccupied states,
which is often connected to particle-hole symmetry breaking. This is consistent with the
particle-hole symmetry breaking upon the pseudogap opening observed in the momentum
structure (Fig. 1n). All above observations can be robustly reproduced and no sample
degeneration was found during the measurement (Supplementary Figs. 4-6).
The opening of a pseudogap below T* that breaks the particle-hole symmetry is
accompanied with several anomalous temperature-dependent behaviours. When
considering the spectral lineshape evolution with temperature based on a simple textbook
picture for a metal, one expects a broadening of the spectra with increasing temperature due
to the increased phase space available for inelastic electron scattering. Further, the opening
of a superconducting energy gap at low temperatures generally sharpens quasi-particles by
suppressing density of states for inelastic electron scattering. In stark contrast, the observed
spectral evolution below T* goes to the opposite direction, suggesting an unusual
broadening mechanism that sets in the pseudogap state. In addition, the spectra no longer
retain a Lorentzian line shape below T*. Our data also provide important information
missed in earlier studies. The pseudogap has previously been reported to disappear in a
subtle manner, manifested as a filling-in of the gapped states in the vicinity of EF (a few
kBT) rather than a clear gap closing 9-11. Different from those previous reports, our results
uncover the pseudogap closing with a clearly shifting energy scale (Fig. 2c) and a
concurrent spectral weight redistribution over a wide energy range (Fig. 2d).
Our observations collectively reveal a rich physical picture of the pseudogap
phenomena beyond a simple extension of superconductivity which necessitates other
ingredients for a complete picture. This picture is overall consistent and aligned with Ref.
10, but differs in critical areas such as the assumption of particle-hole symmetry near the
antinode. Our finding of a particle-hole symmetry breaking supports the idea that the
pseudogap state competes with superconductivity. Motivated by the observations 2-19, 22-25
as well as theoretical implications 26 of spatial symmetry breaking (density-wave-like
correlations), we explore whether a density wave picture can account for the observation.
We first consider a conventional model 27 of long-range density-wave orders in the weak-
coupling limit. Fig.3 shows the expected ARPES spectra from a calculation with
incommensurate checkerboard density-wave order of orthogonal wave vectors, [0.26π, 0] &
[0, 0.26π] (3a) and the commensurate [π, π] density-wave order (3b) for three different
order parameters V = 15, 30 and 60 meV (weighted blue curves from top to bottom). We
find that the modelling can qualitatively reproduce some critical aspects of our findings.
Most importantly, there is no dispersion anomaly at kF because the gap by these density-
wave correlations opens by band-folding in a particle-hole asymmetric manner, which is
very different from the gap opening by superconductivity. Rather, there exists back-
bending (arrows) markedly away from kF (red dashed lines). Besides, these models provide
a qualitatively better account for the dramatic shifting energy scale with temperature than
the existence of homogeneous superconductivity. The most striking is the downward shift
of the band bottom energy (note the similarity between 1n and 3a) which hardly happens
for superconductivity. The modelling, on the other hand, fails to describe the
counterintuitive broadening of spectra with decreasing temperature observed in the present
study. A model assuming a short-range nature (10 lattice constants) of the [π, π] density-
wave order 28, 29 can do a better job regarding the broader spectra in the gapped state (Fig.
3c). In this model, interestingly, shorter density-wave correlation length is required to
reproduce the broader spectra at lower temperature. However, it is difficult to explain the
wide energy range of spectral weight distribution in the pseudogap state given the relatively
low T* with this model. This suggests that the textbook starting point assuming extended
Bloch waves becomes no longer accurate 8. An unconventional explanation that captures
the good aspects of the simple models of competing order but with emphasis on effects of
strong coupling thus seems to be necessary for our observations of the pseudogap. The
highly localized nature indicated by the broadness of the spectra is consistent with this idea.
Reminiscent of the ARPES effects observed here, a particular form of spatial and
particle-hole symmetry broken spectra is observed by momentum-integrated STM in the
pseudogap phase at low temperature2, 3, 30. The observed nanoscale inhomogeneity
associated with local density-wave order and dominated by high-energy states away from
the EF is formally consistent with our spatially-averaged ARPES observation, both pointing
to the symmetry-broken nature and high-energy relevance of the pseudogap phase. The
momentum and energy resolution of ARPES allows us to directly reveal that not only the
spectral weight, as found in STM experiments, but also the gap form itself has a particle-
hole symmetry broken nature in the pseudogap state. Furthermore, the detailed temperature
dependent data made possible by ARPES reveal: i) the strikingly simple electronic structure
of the true normal state above T* - a critical baseline to understand the pseudogap state that
has not been experimentally established; and ii) a direct connection to the symmetry broken
state below T* with the opening of the pseudogap. As such, our finding allows an
integrated picture to advance the understanding of the pseudogap as a strong coupling form
of broken-symmetry state that emerges from a simple normal state above T* and most
likely competes with superconductivity.
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Acknowledgements We thank W.-S. Lee, E. Berg, K. K. Gomes, B. Moritz, S. A. Kivelson, M. Grilli, H. Q.
Lin, N. Nagaosa, A. Fujimori and J. Zaanen for helpful discussions and Y. Li for experimental assistance on
SQUID measurements. R.-H.H. thanks the SGF for financial support. This work is supported by the
Department of Energy, Office of Basic Energy Science under contract DE-AC02-76SF00515.
Author contributions ARPES measurements were done by M.H., R.-H.H., K.T. and W.M. Y.Y. and H.E.
grew and prepared the samples. M.H. and R.-H.H. analyzed the ARPES data and wrote the paper with
suggestions and comments by J.P.T., H.Y, T.P.D. and Z.-X. S. ARPES simulations were done by M.H., R.-
H.H., J.P.T., H.Y. and T.P.D. R.G.M. and D.H.L. maintained the ARPES endstation. Z.H. and Z.-X.S. are
responsible for project direction, planning and infrastructure.
Author Information The authors declare no competing financial interests. Correspondence and requests for
materials should be addressed to Z.-X.S (firstname.lastname@example.org).
Figure 1 Particle-hole symmetry breaking in the antinodal dispersion of
pseudogapped Pb-Bi2201. Tｃ = 34 K, T* = 125 ± 10 K. a-l, Fermi-Dirac function
spectra are smaller than the symbol size. An additional shoulder feature of weak
(FD) divided image plots (upper panels) and corresponding spectra as a function of
parallel momentum (lower panels) taken along the antinodal cut shown in the inset
of g at selected temperatures. The intensity maximum of each spectrum is marked
by circle. Typical error bars estimated in analysis of the smoothed first-derivative
dispersion is seen close to EF at 10 K. Note that this feature at low energy has
been a major focus in previous studies 9-18. Spectra in red and green are at kF and
possible back-bending momenta of the intensity maximum dispersion, respectively
(Supplementary Figs. 1 and 2). m, Simulated dispersion for d-wave homogeneous
superconductivity with order parameter V = 30 meV. The quasiparticle energy for a
given momentum state k is given by
. Cuts are along (π, -π)-
(π, 0)-(π, π). The red (blue) curve is for the true normal (gapped) state. Spectral
weight is indicated by the curve thickness. The back-bending (or saturation) of the
dispersion and kF are indicated in the panel. Note that the back-bending
momentum in the gapped state remains aligned with kF. See details in
Supplementary Method. n, Summary of the intensity maximum dispersions at
different temperatures. Typical error bars from the derivative analysis are smaller
than the symbol size.
Figure 2 Temperature evolution of the antinodal spectra tied to the
pseudogap opening. a, b, FD-divided spectra at (π, 0) and kF, respectively, with
constant vertical offset between each spectrum after the normalization at the
highest binding energy, taken at different temperatures as color-coded in
increments of 10 K. The intensity maxima at 160 K and 10 K are marked by circles.
Note that the low-energy shoulder feature loses its clear definition roughly above Tc
upon raising temperature. The offset and normalization are for better visualization
of the spectral line shape and the intensity maximum feature but not for the
discussion of the spectral weight redistribution (See Supplementary Fig. 3 for
photon flux normalization). c, Temperature-dependent energy position of the
intensity maximum at (π, 0) and two kF along the antinodal cut. Error bars are
estimated in analysis of the smoothed first-derivative spectra. d, Temperature
dependence of the average intensity within [-0.36, -0.30] eV after spectral weight
normalization by the photon flux (I0) and within selected energy ranges as denoted,
of the raw and FD-divided spectra at two kF (see Supplementary Fig. 3 for the
normalized spectra). The average intensity is normalized at 160 K for comparison
with its typical error bars (± 5% for I0 normalization and smaller than the symbol
size for the area normalization). Note that any finite temperature dependence
indicates spectral weight redistribution beyond the window.
Figure 3 Simulated spectra for simple options of the density-wave
correlations behind the pseudogap. Cuts are along (π, -π)-(π, 0)-(π, π) for all
case. Two long-range orders: a, incommensurate checkerboard density-wave
order of orthogonal wave vectors, [0.26π, 0] & [0, 0.26π] and b, commensurate [π,
π] density-wave order. The green dashed curves are shadow bands due to the
corresponding order which interact with the bare band (red) producing the
renormalized band dispersions (blue) presumably observable in the ordered or
locally-ordered state. Vertical red dotted lines are eye guides for kF. Selected
renormalized band dispersions for the same eigenstate with order parameter V =
15, 30 and 60 meV (from top to bottom), independent of k, are shown for each
case to exemplify the misalignment between kF and the back-bending (or saddle
point) momentum (arrow). Spectral weight is indicated by the curve thickness. Note
that only first-order band folding are considered for simplicity in simulations for a.
See Supplementary Fig. 7 for complete results of all long-range density-wave
simulations. c, The [π, π] density wave order with finite correlation length: 10 lattice
constants, and density wave gap ∆k = 60 meV, independent of k. The extracted
dispersion is shown in blue curve with its thickness proportional to the peak
intensity. For a and b, note that a pronounced upshift of the band bottom
commonly requires a decreasing V which inevitably results in corresponding
changes of the back-bending position, in contrast to the “pinning” of it possibly
observed below ~ 50 K (Fig. 1n). This could be consistent with the reported
temperature independence of back-bending momenta across Tc 12, and poses a
challenge for the exclusive density-wave picture put in such simple ways. See
details in Supplementary Method.
Supplementary Information for
Particle-Hole Symmetry Breaking in the Pseudogap State
M. Hashimoto*1, 2, 3, R.-H. He*1, 2, K. Tanaka1, 2, 3, 4, J. P. Testaud1, 2, 3, W. Meevasana1, 2, R.
G. Moore1, 2, D. H. Lu1, 2, H. Yao1, Y. Yoshida5, H. Eisaki5, T. P. Devereaux1, 2, Z.
Hussain3, & Z.-X. Shen1, 2
*These authors contributed equally to this work.
Samples and Experimental Method
Nearly optimally-doped high-quality single crystals of Pb0.55Bi1.5Sr1.6La0.4CuO6+δ
(Pb-Bi2201) were grown by the travelling solvent floating-zone method. The Pb
doping suppresses the super-modulation in the BiO plane and minimizes
complications in the electronic structure due to photoelectron diffraction. This
aspect, together with the absence of bilayer splitting, allows a clean tracking of the
spectral evolution of pseudogapped antinodal states. The carrier concentrations of
the samples were carefully adjusted by a post-annealing procedure under flowing N2
gas which varies the oxygen content. All data shown were obtained on samples at the
same doping level from the same batch of growth and similar post-annealings. The
estimated hole concentration from the Fermi surface area is ~25 % which is
consistent with previous reports for near optimal doping 31, 32. The onset temperature
of superconducting transition, Tｃ, determined by SQUID magnetometry, was 34 K
with a transition width less than 3 K. X-ray and Laue diffraction showed no trace of
impurity phases. The moderate size of T* (125 K ± 10 K) allows reference data to be
obtained in the true normal state without involving severe thermal smearing or
causing the degradation of cleaved sample surfaces (Figs. S4-S6), which would
become a problem for lower dopings with higher T* and thus requiring higher
measurement temperatures. Compared with higher Tc cuprates, the sufficiently large
difference between T* and Tc benefits a systematic study of the pseudogap physics
in a wide temperature range without entanglement of coherent superconductivity.
ARPES measurements were performed at beamline 5-4 of the Stanford Synchrotron
Radiation Laboratory (SSRL) with a SCIENTA R4000 electron analyzer. All data shown in
this paper were taken using 22.7 eV photons in the first (mainly) and second (partially)
Brillouin zones with total energy and angular (momentum) resolutions of ~ 10 meV and ~
0.25° (~ 0.0096 Ǻ-1), respectively. The temperatures were recorded closest to the sample
surface position within an accuracy ±2 K. The samples were cleaved in situ at various
temperatures ranging from 10 K to 160 K and measured in an ultra high vacuum chamber
with a base pressure of better than 3×10-11 Torr which was maintained below 5×10-11 Torr
during the temperature cycling. Measurements on each sample were completed within 48
hours after cleaving.
To remove the effect due to cutoff by the Fermi-Dirac (FD) function, raw ARPES spectra
(e.g., Fig. S3c and d) have been divided by a convolution of the FD function at the given
temperature and a Gaussian representing energy resolution. This procedure allows us to
recover the actual band dispersions closest to EF and trace them above EF, where thermal
population leads to appreciable spectral weight especially at high temperatures. Its effect on
the spectral line shape is limited to ~ 4kBT around EF and our analysis of the dispersion of
the maximum in spectral intensity at high energy remains robust (Fig. S2). We avoid
discussing the intensity above EF where the resolution effects become stronger 13. We used
this method for the energy-distribution-curve (EDC) analysis rather than an EF
symmetrisation procedure 9-11, 16, 17, 33 because the latter implicitly assumes the particle-hole
symmetry of states which is shown to be broken at kF in this study. By going above T*, this
analysis allows us to objectively determine kF, which was often assumed to be the
dispersion back-bending momentum in the gapped state (so called minimum gap locus) 12.
As we show, the discrepancy between these two momenta remains finite within a wide
temperature range below T*.
Note here that the spectra in Fig.S3 were normalized by two methods, photon flux which
gives the raw spectral weigh, or normalization within a certain energy window. The second
method is used as a supplement to double check whether spectral weight is conserved
within a certain energy window. If spectral weight in not conserved within the
normalization window, then one should see a change outside this window of normalization.
This provides a supplementary mean to check the energy scale of spectral weight
We describe in Main Figs. 3 the modification of the band structure due to the existence of
some density wave order of a wave vector q based on two simple approaches, with one
assuming a long-range order 27, 34 and the other taking into account short-range fluctuation
effects 28, 29. For the first case, the mean-field Hamiltonian is given by:
where Vq is the interaction (order parameter)
between the main band at k and the shadow band at k+q generally as a result of coupling of
electron to some bosonic mode (e.g., phonons for a charge density wave) of a wave vector
k c (
k c) is the creation (annihilation) operators for electrons at k.
k εis the tight-binding
bare band dispersion which is obtained by a global fit to the experimental low-energy band
dispersions at 160 K. In case of the commensurate q = qAF = [π, π], the eigenstate
for the Hamiltonian with the eigenenergy
obtained by solving the matrix
. In Main Fig. 3a and b and Fig. S7, the
blue curve shows the renormalized band dispersion
with its thickness proportional to
. In the incommensurate checkerboard case with q1 = [0.26π, 0] and
q2 = [0, 0.26π],
. Here we truncate the interaction at
the first order and only take into account the first-order shadow bands in order to show a
simple and clear physical picture. The qualitative existence of dispersive states at high
energy and dispersion back-bendings (or saddle points) apart from kF remains robust in
different calculations which apply different schemes for the truncation. Complications due
to the weak but non-vanishing influence by higher orders, nq1 and nq2 can produce more
folded bands of different spectral weight at a given momentum that can in principle
comprise a broad spectral envelope reminiscent of the experimental observation. However,
we note that the weak high-order constituents might be practically undetectable 34 without
an anomalous suppression of the main band. The superconductivity case shown in Fig. 1m
is based on the Hamiltonian
where a d-wave superconducting gap is assumed
2 / )
, with the
kv are the BCS coherence factors.
We consider density wave fluctuations in the spirit of Refs. 28 and 29. Specifically, we
assume a spectral function
, where the Green’s function obeys a
interacting Green’s function and
representing a Lorentzian distribution of ordering
wave vectors having a width associated with the finite correlation length. By choosing an
ordering wave vector of [π, π], the resulting spectral function is shown in Main Fig. 3c.
We note, for simplicity, we chose not to include the cases of pair density wave. Such
options should also be explored in a more comprehensive theory.
Figure S1 Possible back-bending of the intensity maximum at 40 K. a, Image plot of the
FD-divided ARPES spectra. Red and blue squares denote the momenta of kF at 160 K and
those of possible back-bendings at 40 K, respectively. b, EDCs in the red square region as
indicated in a. The color EDCs correspond to the momenta as indicated by the squares. The
intensity maxima of EDC are marked by blue circles, same as shown in a, showing the
dispersion flatters at low energy where the back-bending position is defined. To estimate
the intensity maxima, we first applied first derivative to the FD-divided spectra and found
peak of the derivative spectra after applying moderate smoothing to eliminate the noise. We
defined the maxima from the extrapolation of the smoothing-degree dependence of the peak
position to zero smoothing.
Figure S2 Robustness of the dispersion of intensity maximum against the FD-division and
background subtraction. The same data set at 50 K is used as in Main Fig. 1j. a, FD-divided
spectra. b, Background (BG) subtracted FD-divided spectra. c, Raw spectra. Intensity
maximum positions determined from the smoothed first-derivative analysis are plotted as
circles in a-c. Spectra in red and green are at kF and possible back-bending momenta of the
intensity maximum dispersion, respectively. Blue spectrum in a is a FD-divided spectrum
at far from kF in the same cut, which is used for the background subtraction. d, Summary of
the dispersions. Note that the difference between different analyses is within error bars.
Figure S3 Normalization dependence of the temperature evolution of FD-divided and
raw EDCs at kF. a,b are for the FD-divided spectra and c,d for the raw spectra taken at
different temperatures as color coded. The same set of data as shown in Main Fig. 2b is
used. The energy windows for normalization of a and c are [-0.30, 0] and [-0.30, 0.09] eV,
respectively. b and d are for the normalization by experimental photon flux. The average
intensity within [-0.36, -0.30] eV is plotted in Main Fig. 2d as a function of temperature.
Figure S4 Reproducibility of the antinodal spectra on different samples with three
different sample cleavings at the two selected temperatures. a, b, Image plots of the FD-
divided spectra approximately along (π, -π)–(π, 0)–(π, π) at 160 K and 10 K with two
different flesh cleavings at 160 K and 10 K, respectively. c, d, Selected EDCs from a and b,
respectively. The intensity maximum at 160 K and 10 K are marked by red and blue circles,
respectively. Their traceable dispersions along the cut are shown correspondingly in a and
b. The dispersions obtained from another cleaving whose data are shown in the main text
are overlaid (light blue circles) in a and b for comparison. Note that the cut locations are
close but not exactly the same for different data sets. The shoulder features near EF (Main
Fig. 1l) are also seen in d.
Figure S5 Reproducibility of the ungapped Fermi surface shape at 160 K on two
samples with different sample cleaving temperatures. Data shown in the main text were
taken on the sample cleaved at 10 K. The Fermi surface map at 160 K was obtained after
completing the temperature dependence study from 10 K to 160 K. The spectral weight
contour (EF ± 10 meV) at 160 K on another sample fleshly cleaved at 160 K with identical
experimental conditions is shown here. kF determined from the momentum distribution
curve (MDC) at EF for the two data sets are overlaid for comparison. The inset shows the
MDCs at EF approximately along (π, -π)–(π, 0)–(π, π) for the two data sets. Blue squares
and dashed lines are guides to eyes showing kF for the cleaving at 10 K. The shape of Fermi
surface is representative of samples from the same batch at the same doping level,
suggesting that no sample aging was involved in the temperature dependence study.
Figure S6 Nodal dispersions without the pseudogap complication before and after the
same temperature dependence study presented in the main text. MDC dispersions
along the cut shown in a at 10 K (blue) and 160 K (red) are plotted together in b, showing
no detectable chemical potential shift or sample aging.
Figure S7 Complete results of the long-range density-wave modelling. a-d and e-h are
for Main Fig. 3a and 3b, respectively. a and e show bare and shadow bands as specified by
the corresponding eigenvector, which mutually interact to form renormalized band
dispersions shown in b-d and f-h for different V. Arrows point out bands of the same
eigenstates after interaction that are shown in Main Fig. 3a and 3c.
3 Download full-text
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