Pseudogap and Fermi surface topology in the two-dimensional Hubbard model

Wei Wu,1, 2 Mathias S. Scheurer,3 Shubhayu Chatterjee,3 SubirSachdev,3, 4, 5 Antoine Georges,2, 6, 1, 7 and Michel Ferrero1, 2

1Centre de Physique Theorique, Ecole Polytechnique,CNRS, Universite Paris-Saclay, 91128 Palaiseau, France

2College de France, 11 place Marcelin Berthelot, 75005 Paris, France3Department of Physics, Harvard University, Cambridge MA 02138, USA

4Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y55Department of Physics, Stanford University, Stanford, CA 94305

6Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth avenue, New York, NY 10010, USA7DQMP, Universite de Geneve, 24 quai Ernest Ansermet, CH-1211 Geneve, Suisse

One of the distinctive features of hole-doped cuprate superconductors is the onset of a ‘pseudogap’below a temperature T ∗. Recent experiments suggest that there may be a connection between theexistence of the pseudogap and the topology of the Fermi surface. Here, we address this issueby studying the two-dimensional Hubbard model with two distinct numerical methods. We findthat the pseudogap only exists when the Fermi surface is hole-like and that, for a broad rangeof parameters, its opening is concomitant with a Fermi surface topology change from electron-to hole-like. We identify a common link between these observations: the pole-like feature of theelectronic self-energy associated with the formation of the pseudogap is found to also control thedegree of particle-hole asymmetry, and hence the Fermi surface topology transition. We interpretour results in the framework of an SU(2) gauge theory of fluctuating antiferromagnetism. We showthat a mean-field treatment of this theory in a metallic state with U(1) topological order providesan explanation of this pole-like feature, and a good description of our numerical results. We discussthe relevance of our results to experiments on cuprates

I. INTRODUCTION

A very debated topic in the physics of high-temperature superconductors is the nature of the ’pseu-dogap’1,2 in their phase diagram. Below a tempera-ture T ∗(p) which is a decreasing function of the hole-doping level p, a pseudogap develops, correspondingto a suppression of low-energy excitations apparentin many experimental probes. Extrapolated to zero-temperature, T ∗(p) defines a critical hole doping p∗

above which the pseudogap disappears as doping is in-creased. Another important critical value of the dop-ing, denoted here pFS, is that at which the Fermi sur-face topology changes from hole-like to electron-like,corresponding to a Lifshitz transition. Recent exper-iments on Bi2Sr2CaCu2O8+δ (Bi2212) have suggestedthat the pseudogap may be very sensitive to the Fermisurface (FS) topology and that p∗ ' pFS in this com-pound3,4. In a simultaneous and independent mannerfrom the present theoretical work, Doiron-Leyraud et al.5

recently performed a systematic experimental study us-ing hydrostatic pressure as a control parameter in theLa1.6−xNd0.4SrxCuO4 (Nd-LSCO) system, and an unam-biguous connection between FS topology and the pseu-dogap was found.

In this work, we investigate this interplay by study-ing the two-dimensional Hubbard model. In the weak-coupling scenarios of pseudogap physics, there is a nat-ural connection between the FS topology and the co-herence of low-energy quasiparticles. Indeed, for a hole-like FS, coherence is suppressed at the ‘hot spots’ wherethe FS intersects the antiferromagnetic zone boundary.

When the FS turns electron-like, increased quasiparticlecoherence is restored all along the FS (see Appendix fora more detailed analysis)6–11. At stronger coupling, sev-eral methods9,12–28 have established that the Hubbardmodel displays a pseudogap which originates from an-tiferromagnetic correlations. These correlations becomeshort-range as the coupling strength or doping level areincreased, as found in experiments29. The FS topology,on the other hand, is an issue which has to do with low-energy, long-distance physics. Hence, it is an intrigu-ing and fundamentally important question to understandhow the short-range correlations responsible for the pseu-dogap can be sensitive to FS changes.

Here, we study the Hubbard model for a broad rangeof parameters, and analyze the pseudogap and Fermisurface topology, and their interplay. We show that,at strong coupling, interactions can strongly modify theFermi surface, making it more hole-like as compared toits non-interacting shape16,30–33. We find that a pseudo-gap only exists when the FS is hole-like, so that p∗ ≤ pFS.We identify an extended parameter regime in which thesetwo critical doping levels are very close to one another:p∗ ' pFS, so that the FS turns electron-like only whenthe pseudogap collapses. Moreover we show that, whenconsidering the relation between the pseudogap and FStopology, hole-doped cuprates can be separated into twofamilies: materials for which p∗ ' pFS and materialswhich have p∗ < pFS . These two families differ mostlyby the relative magnitude of the next nearest-neighborhopping. These findings are shown to be consistent witha large body of experiments on cuprates.

We reveal that a common link between these observa-

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tions is the pole-like feature22,23,30,32,34–36 displayed bythe electronic self-energy at the antinodal point, k =(π, 0). The large imaginary part of the antinodal self-energy associated with this pole is responsible for thepseudogap, while the large particle-hole asymmetry asso-ciated with its real part controls the interaction-induceddeformation of the Fermi surface and the location of theFermi surface topology transition. We investigate theevolution of this particle-hole asymmetry as a functionof doping and nearest-neighbor hopping t′, and showthat the line in (p, t′) space where particle-hole symme-try is approximately obeyed at low energy is pushed, atstrong coupling, to very low values of p and very nega-tive values of t′. This is in stark contrast to the results ofweak-coupling theories where this line is close to the Lif-shitz transition of the non-interacting system. This alsoexplains why interactions drive the Fermi surface morehole-like for hole-doping.

In order to understand these results from a more ana-lytic standpoint, we consider a recently developed SU(2)gauge theory of fluctuating antiferromagnetic order37,38;additional results on the SU(2) gauge theory appear in acompanion paper, Ref. 39. We focus on a metallic phaseof this theory, characterized by U(1) topological order,which does not break spin or translational symmetries.We show that a mean-field treatment of this gauge the-ory provides a good description of our numerical results.In particular, the self-energy of the charge-carrying field(chargon) in this theory displays a pole which providesan explanation for the quasi-pole of the physical electronself-energy. The latter is calculated and compares well toour numerical results, as do the trends in the evolution ofthe pseudogap and particle-hole asymmetry as a functionof p and t′.

This paper is organized as follows. In Sec. II, we brieflyintroduce the model and the numerical methods used inthis article. In Sec. III, we study the interplay betweenthe pseudogap and FS topology and analyze the mech-anisms controlling this interplay. The comparison andinterpretation of our results in terms of the SU(2) gaugetheory is presented at the end of this section. In Sec.IV we discuss the relevance of our results to experimentson hole-doped cuprates. Sec.V provides a conclusion andoutlook. Finally, details about the employed methodsand various supporting materials can be found in theAppendices.

II. MODEL AND METHOD

We consider the Hubbard model defined by the Hamil-tonian:

H = −∑ij,σ

tijc†i,σcj,σ + U

∑i

ni↑ni↓ − µ∑i,σ

niσ, (1)

where U is the onsite Coulomb repulsion and µ the chem-ical potential. The hopping amplitudes tij ’s are chosento be non-zero between nearest-neighbor sites (tij = t)

and next-nearest-neighbor ones (tij = t′). These hop-ping amplitudes define a non-interacting dispersion re-lation εk = −2t(cos kx + cos ky) − 4t′ cos kx cos ky. Inthe following, t = 1 will be our unit of energy. Wesolve this model using two distinct methods: the dy-namical cluster approximation (DCA15) and determinantquantum Monte Carlo (DQMC40), see the Appendixfor details. Cluster extensions of dynamical mean-fieldtheory (DMFT) have shown that the Hubbard modelis able to capture many features of cuprate supercon-ductors, such as the superconducting dome and thepseudogap9,14–18,20–23,26,32,41–43. They have also estab-lished that the pseudogap originates from antiferromag-netic correlations, which become short-range as the cou-pling strength or doping level are increased. This wasalso recently corroborated by exact diagrammatic MonteCarlo simulations28. While cluster extensions of DMFThave shown that hole doping can drive a Lifshitz tran-sition30–32 no general relationship between the pseudo-gap and FS topology has been established. We thereforecarry out a systematic study for a broad range of param-eters in order to investigate this issue.

III. RESULTS

A. Pseudogap and Fermi surface topology

In Fig. 1, we display the pseudogap onset tempera-ture T ∗(p), and the temperature TFS(p) at which theFermi surface changes its topology, as a function of dop-ing level p, for several values of the next-nearest-neighborhopping t′. T ∗ is identified as follows: we calculate thezero-frequency extrapolated value of the spectral func-tion at the antinodal point (π, 0); we find that its tem-perature dependence displays a maximum which we iden-tify as T ∗. Below this scale, the antinodal spectral in-tensity decreases, signaling the opening of a pseudogap.TFS is identified as the temperature where the Fermi sur-face crosses the (π, 0) point, and turns from hole-like toelectron-like as temperature decreases (see below). Notethat our definition of a Fermi surface is a pragmatic one:strictly speaking a Fermi surface only exists at zero tem-perature. At finite temperatures, we define the Fermisurface as the surface in momentum-space correspondingto the maximum of the spectral intensity as it wouldbe observed, e.g. in an angle-resolved photoemission(ARPES) experiment44.

When extrapolated to zero temperature, these datadefine two critical doping levels: p∗ such that the pseu-dogap disappears for p > p∗, and pFS that marks thetransition from a hole-like FS (p < pFS) to an electron-like FS (p > pFS). Strikingly, the two curves in Fig. 1suggest that the pseudogap can only exist when the Fermisurface is hole-like, i.e. that p∗ ≤ pFS. It appears that forvalues of t′ ≥ −0.1 both transitions happen at the samedoping p∗ = pFS within our error bars. For more negativevalues of t′ the Fermi surface first becomes hole-like as p

3

(c)(b)(a)

Hole-like FS

Electron-like FS

Figure 1. Pseudogap and Lifshitz transition temperatures. Evolution of the pseudogap onset temperature T ∗ (black)and the Lifshitz transition temperature TFS (blue) as a function of the hole doping p, for several values of t′ and U = 7. Thefinite temperature data points are extrapolated to zero temperature and yield two critical dopings p∗ and pFS. It is apparentthat p∗ ' pFS for t′ = 0 and t′ = −0.1, while p∗ < pFS for t′ = −0.2. The solid lines are linear (for T ∗) and quadratic (for TFS)least squares fits to the data points, except the TFS line of t′ = 0 where TFS collapses to zero close to p∗ . Error bars estimateall uncertainties in finding T ∗ and TFS with DCA (see also Appendix). Note that the change of topology of the Fermi surfacefor the interacting system occurs at a larger doping than that of the non-interacting system (indicated by a light-blue arrow).

is reduced, and the pseudogap opens at a lower doping,i.e. p∗ < pFS. We never observe a pseudogap with anelectron-like Fermi surface, which would correspond top∗ > pFS.

This can be documented further by repeating this anal-ysis for several doping levels p and t′ values. The result-ing map in the (p, t′) parameter space is displayed inFig. 2. A first observation is that the topological tran-sition of the FS (blue line) that separates the regionswith hole-like and electron-like Fermi surfaces is stronglyrenormalized with respect to its non-interacting (U = 0)location (dashed line in Fig. 2 and arrows in Fig. 1). Theblack line defines the onset of the pseudogap. These linesdefine three regions: at large doping above the blue line,the FS is electron-like and no pseudogap is present. Inthe intermediate region between the two lines, the FSis hole-like but without a pseudogap. The topologicaltransition and pseudogap opening coincide for a rangeof t′, while for more negative t′ the two lines split apartand, as doping is reduced, the pseudogap only opens af-ter the FS has already turned hole-like at higher dopinglevel (p∗ < pFS). The pseudogap and FS topology tran-sition lines are dependent on the value of U . As detailedin the Appendix, a larger value of U yields a more ex-tended regime of parameters for which p∗ ' pFS, withthe ‘branching point’ where the two lines merge mov-ing towards more negative values of t′ and larger dopinglevel. This observation is important when comparing toexperimental observations (see below).

Figure 2. Zero-temperature Fermi surface topologyand pseudogap in the p−t′ plane. The black line separatesa region with no pseudogap (no PG) from a region wherea pseudogap exists (PG). The blue line indicates where theinteracting Fermi surface changes its topology from electron-like (e-FS) to hole-like (h-FS). The dashed blue line signalsthe same transition in the non-interacting case. The red curvelocates the change in particle-hole asymmetry at the antinode:above the red-line the real-part of the self-energy modifies theFS towards a more hole-like shape. On the red line, the self-energy pole crosses zero-energy and approximate particle-holesymmetry is restored, corresponding also to a maximum of thelow-energy scattering rate as t′ is varied for fixed p. PointsA-D label a set of parameters which are discussed further inFig. 4.

4

(a) (b)

Figure 3. Antinodal quasiparticle dispersion andFermi surface topology (a) Antinodal quasiparticle energyε(π,0) for different doping levels, as a function of temperature.The pseudogap onset temperature T ∗ and the Lifshitz tran-sition temperature TFS are indicated by arrows. Below thepseudogap temperature, ε(π,0) rapidly becomes very negative,driving the FS hole-like. Only when no pseudogap is present(here for p > 0.12) does ε(π,0) increase at low temperature andeventually becomes positive to yield an electron-like Fermisurface. (b) Illustration of the relation between the sign ofε(π,0) and the Fermi surface topology.

B. Change of Fermi surface topology due tocorrelation effects

The Fermi surface topology at the antinode is con-trolled by the renormalized quasiparticle energy

ε(π,0) = ε(π,0) − µ+ ReΣ(π,0)(ω = 0)

= 4t′ − µ+ ReΣ(π,0)(ω = 0) (2)

For negative values of ε(π,0) the Fermi surface is hole-like, while it is electron-like for ε(π,0) > 0. In orderto gain insight in the mechanisms driving the Lifshitztransition, Fig. 3 displays ε(π,0) as a function of temper-ature for various doping levels, with arrows indicatingT ∗ and TFS. Interestingly, even at the highest temper-ature T = 0.2 displayed there, ε(π,0) is negative for alldoping levels, yielding a hole-like Fermi surface while thenon-interacting Fermi surface would be electron-like forp & 9%. This temperature is above the pseudogap tem-perature T ∗, and hence the renormalization of the FSwould be visible on a full Fermi surface in an ARPESexperiment. In this high-temperature range, only localcorrelations are responsible for this effect, as already cap-tured in a single-site DMFT calculation (see Fig. 12 in theAppendix). As temperature is decreased, ε(π,0) first in-creases slightly but then suddenly drops to very negativevalues, pushing the Fermi surface to be very hole-like atlow temperatures. This starts happening just above thepseudogap temperature and both effects can be tracedback to non-local electronic correlations. For this valueof t′ = −0.1 the connection between the disappearance ofthe pseudogap and the recovery of an electron-like sur-face is clear. Indeed, when no pseudogap is present ase.g. for p = 0.15, ε(π,0) keeps on increasing and crosseszero, and an electron-like FS is recovered at low-T .

C. Particle-hole asymmetry and pole-like structurein the self-energy

From the definition of ε(π,0) it is clear that it is the realpart of the self-energy at the antinode that drives therenormalization of the FS. In Fig. 4a, we consider a fixed

doping level p = 5% and display ReΣ(2)(π,0)(ω = 0) as a

function of t′, in which Σ(2) ≡ Σ−Up/2 is the self-energyfrom which the Hartree (infinite frequency) contribution

has been subtracted out. It is seen that ReΣ(2)(π,0)(ω = 0)

changes sign around t′ ' −0.2 and becomes negative andfairly large for larger values of t′. This pushes the Fermisurface topology transition to higher values of t′: for 5%doping it remains hole-like up to t′ ' +0.2 whereas theLifshitz transition of the non-interacting system occursat t′ ' −0.05 (see also Fig. 2).

The real part of the self-energy is related to its imagi-nary part through the Kramers-Kronig relation

ReΣ(2)k (ω = 0) =

1

π

∫ ∞0+

ImΣk(ω′)− ImΣk(−ω′)ω′

dω′.

(3)It is therefore instructive to analyze the behavior ofImΣ(π,0)(ω) (Fig. 4b) for several values of t′ (as indi-cated by the points A, B, C and D on Fig. 2) corre-sponding to positive, vanishing and negative values of

ReΣ(2)(π,0)(ω = 0). In all four cases, the imaginary part of

the self-energy displays a prominent peak, correspondingto a pole-like feature of the self-energy. For t′ = −0.2(point B), this peak is centered at ω = 0. Because itis particle-hole symmetric, it leads to a vanishing realpart of the self-energy (see Fig. 4a). For values of t′

just below and above -0.2 (points A and C), the peak inImΣ(π,0)(ω) shifts to negative (resp. positive) values ofω. It has become particle-hole asymmetric and inducesa positive (resp. negative) real part of the self-energy.There is therefore a direct connection between the ex-istence of a large particle-hole asymmetric peak in theimaginary part of the self-energy and the renormaliza-tion of the Fermi surface to a more hole-like topology.Note that the largest value of the low-frequency scat-tering rate as t′ is varied is found when ImΣ(π,0)(ω) isparticle-hole symmetric (e.g. point B in Fig. 4): this de-fines the location of the red line in Fig. 2 (see also theAppendix). Anywhere above this line, the self-energyis particle-hole asymmetric and drives the Fermi surfacetopology transition to larger doping p as compared to thenon-interacting case. Note that the system becomes veryincoherent below the red line, at more negative valuesof t′ and small doping. The precise nature of the Fermisurface in this region, and its possible reconstruction, isdifficult to assess with the methods employed here.

This pole-like feature in the self-energy is also respon-sible for opening the pseudogap, as clearly seen fromthe inset of Fig. 4b which displays the antinodal spec-tral function: the minimum of the spectral intensity isfound to coincide with the frequency of the quasi-pole,

5

(a) (b)

Figure 4. Evolution of the antinodal self-energy at fixed doping p = 0.05, as a function of t′. (a): Real (Hartreesubtracted, see text) and imaginary parts of the antinodal self-energy at ω = 0. The real part vanishes where the imaginarypart is maximum, corresponding to a particle-hole symmetric low-energy ImΣ(π,0)(ω). (b): Real-frequency scattering rateImΣ(π,0)(ω) obtained from the Maximum Entropy method for different values of t′. It displays a pole-like feature that crosseszero at t′ ' −0.2 (point B) where the low-energy scattering is maximum. When the pole is on the positive energy side,it induces a negative real part of the self-energy (through the Kramers-Kronig relation) that drives the Fermi surface morehole-like. Inset: antinodal spectral function at point D at T = 1/30. See Fig. 2 for the locations of points A−D in the (p, t′)plane.

where ImΣ(π,0)(ω) is largest.

D. Fermi surface topology: numerically exactDQMC results

These results have been obtained using the DCA ap-proximation with an 8-site cluster (see Appendix A). Wealso cross-checked these results with a different and inde-pendent method: numerically exact determinant quan-tum Monte Carlo (DQMC)40 at T = 1/3. The resultis displayed in Fig. 5 (left panel) and clearly shows thatthe antinodal self-energy drives the Fermi surface hole-like over a broad region of the (p, t′) plane, in agreementwith our DCA calculations. One can again observe a

line where ReΣ(2)(π,0)(ω) vanishes, mapped out for several

values of U in the right panel. This line compares withthe red line of Fig. 2, and moves closer to half-filling asU is increased (see also Fig. 11) and towards the non-interacting Lifshitz transition line as U is reduced.

E. SU(2) gauge theory

Recent numerical work, using a ‘fluctuation diag-nostics’ analysis of the contributions to the electronicself-energy in both the DCA26 and lattice diagram-matic Monte-Carlo28 approaches have established thatthe pseudogap is associated with the onset of short-range antiferromagnetic (AF) correlations. On the an-alytical side, an SU(2) gauge theory approach has beenintroduced37–39 to deal with states in which AF long-range order is destroyed by orientational fluctuations ofthe order parameter. It is thus very natural to attemptto interpret our numerical results in this framework andcompare them to a mean-field treatment of this gauge

theory.This approach is based on the following representation

of the physical electron fields on each lattice site i:(ci↑ci↓

)= Ri

(ψi+ψi−

)(4)

In this expression, ψ± are ‘chargons’ - fermions whichcarry charge but no spin quantum numbers and Ri’sare 2 × 2 unitary matrix fields, the bosonic spinons

(RiR†i = R†iRi = 1). The Ri matrix can be thought of

as defining the local reference frame associated with thelocal AF order (for early work promoting the local ref-erence frame to a dynamical variable, see Refs. 45–47).This representation has a local gauge invariance corre-

sponding to Ri → RiV†i , ψi → Viψi, with Vi an SU(2)

matrix. The Hubbard interaction can be decoupled us-ing a vector field Φi conjugate to the local spin-density

c†iασαβciβ/2, and a vector ‘Higgs field’ is introduced suchthat:

σ ·Hi = R†iσRi ·Φi . (5)

This identifies the Higgs field, Hi, as the local antiferro-magnetic moment in the rotated reference frame. NotethatHi, which transforms under the adjoint of the gaugeSU(2), does not carry any spin since it is invariant undera global spin rotation.

We can now consider Higgs phases in which 〈Hi〉 6= 0but 〈Ri〉 = 0. Because of the latter, such phases do notdisplay long-range AF order, which has been destroyedby orientational fluctuations. However, 〈Hi〉 6= 0 signalsthat the local order has a non-zero amplitude. A non-zero〈Hi〉 also implies that such a phase has topological order,corresponding to different possible residual gauge groupsonce the SU(2) gauge symmetry has been spontaneouslybroken by the Higgs condensate48–51. There are differentpossible mean-field solutions for the Higgs condensate,

6

U = 5

(2)

= 0(2)

Figure 5. Particle-hole asymmetry from determinant quantum Monte Carlo (see methods). (a): Real part of

the self-energy in the p − t′ plane for U = 5. For a broad region (indicated in red), ReΣ(2)

(π,0) is negative, hence driving the

Fermi surface more hole-like, in agreement with our DCA results. (b): The line where ReΣ(2)

(π,0) = 0 and where the antinodal

scattering rate is largest is indicated for different values of U (to be compared to the red line in Fig. 2). As U is increased theregion where the FS is driven hole-like becomes larger.

corresponding to different topological orders and differ-ent broken discrete symmetries38. Here we shall focus onthe simplest one with U(1) topological order which pre-serves all space group, time-reversal, and spin rotationssymmetries; this corresponds to the following configura-tion of the Higgs field (which resembles AF order):

〈Hi〉 =(0, 0, H0 e

iQ·Ri), (6)

in which H0 is the Higgs field amplitude and Q = (π, π).Solving the gauge theory at the mean-field level, the

Green’s function and self-energy of the chargon field iseasily calculated. Because the chargon field ‘sees’ anantiferromagnetic environment, it is identical to the ex-pression obtained for an antiferromagnetic spin-densitywave39. It thus has a matrix form which involves bothcomponents which are diagonal in momentum and off-diagonal components coupling k to k +Q:

Gψ(ω,k)−1 =

(ω − ξψk H0

H0 ω − ξψk+Q

)(7)

Its momentum diagonal component reads

Gψ(ω,k) =[ω − ξψk − Σψ(ω,k)

]−1Σψ(ω,k) =

H20

ω − ξψk+Q + i0+(8)

In this expression, ξψk = −2Ztt(cos kx + cos ky) −4Zt′t

′ cos kx cos ky − µ is the renormalized dispersion ofthe chargons. A quantitative calculation of the renormal-ization factors Zt and Zt′ requires a full solution of themean-field equations. We found typical values Zt ∼ 0.3and Zt′ ∼ 0.2, weakly dependent on the doping levelp since the chemical potential mainly affects the char-gon dispersion but not the spinon dispersion. Impor-tantly, the self-energy (8) of the chargons has a pole at

ωk = ξψk+Q. Hence the mean-field chargon Green’s func-tion has zeros: these zeros are located at zero energy on

the Brillouin zone contour defined by ξψk+Q = 0, corre-sponding to a chargon ‘Luttinger surface’. There are twobands of chargon excitations, corresponding to the solu-

tions of (ω − ξψk )(ω − ξψk+Q)−H20 = 0. To summarize, a

crucial aspect of this SU(2) gauge theory description isto have chargons whose dispersions are identical (at themean-field level) to the excitations of a spin-density wavestates, despite the theory having no long-range order orbroken symmetries (i.e. the symmetry is restored by thefluctuations of the spinon fields).

At the mean-field level, in the phase associated withthe configuration of the Higgs field considered here, thespinon excitations are gapped. In order to obtain thephysical electron Green’s function, a convolution of thechargon and spinon Green’s function over frequency andmomentum must be performed: Gc = GR ? Gψ and thephysical electron self-energy can then be obtained fromΣ = ω+µ−εk−G−1c (with εk the bare dispersion definedabove). For the purpose of the present paper, a detaileddiscussion of the spinon dispersion and Green’s functionis not essential, see Appendix F and Ref. 39 for details.It is sufficient here to emphasize the two following points.(i) The convolution mainly broadens the pole structureof Gψ but the location in momentum and frequency ofthe most singular structures of the physical self-energyare still those encoded in the chargon self-energy givenby (8). (ii) The convolution does bring an important ef-fect however: in contrast to the imaginary part of thechargon self-energy, which is constant all along the Lut-

tinger surface ξψk+Q = 0, the imaginary part of the phys-ical electron self-energy obtained from the convolution ofGreen’s functions has an imaginary part which is largerclose to the antinodes than close to the nodes, see Fig. 15in Appendix F. Hence, the gauge theory manages to cap-ture qualitative aspects of the nodal-antinodal dichotomyfound in our DCA calculations.

The figure also shows that the peak frequency ωp shiftsfrom negative to positive frequency as t′ is increased. The

7

inset of this figure displays the corresponding spectralfunction at the antinode, which has a pseudogap causedby the quasi-pole at ωp. Note that the pseudogap isparticle-hole asymmetric, as expected from the fact thatit does not originate from the particle-particle channel.These results are in excellent qualitative agreement withthe DCA calculations above (Fig. 4). Note that, for thesake of comparison to the finite-temperature DCA re-sults, the gauge theory calculations presented here areperformed at a finite temperature larger than the spinongap. How do gapless nodal excitations survive in thegauge theory description as temperature is lowered be-low this gap (e.g. by having bound-states of the chargonsand spinon as in an FL* state52) is an important questionwhich is however beyond the scope of the present paper.

In Fig. 6(a), we summarize important aspects of themean-field analysis of the gauge theory39 as a function ofdoping level p and t′. As in Fig. 2, the blue line in thisfigure is the location of the Lifshitz transition of the phys-ical electron FS from hole to electron-like (as defined bythe change of sign of the renormalized antinodal disper-sion, Eq. 2) and the red line indicates where ωp = 0 (i.e.where particle-hole symmetry is approximately restoredat low energy). In good qualitative agreement with theDCA results displayed in Fig. 2, one sees that the Lifshitztransition of the physical FS is pushed to much largerdoping in comparison to that of the non-interacting sys-tem (dashed line), and that the location of the red linewhere the pole is close to zero energy is pushed to muchsmaller doping. The latter approximately coincides withthe Lifshitz transition of the chargon Luttinger surface,given by 4Zt′t

′ = µ. Because the chemical potential µ ofthe interacting system takes more negative values thanthe non-interacting one and also because Zt′ < 1, the redline is shifted to lower doping as U increases, in agree-ment with the result of Fig. 5. This clarifies why thepole is found at positive energies for most values of (p, t′)and why the FS id driven hole-like in a wide region ofthe (p, t′) plane. A striking consequence of the presenceof the pole is illustrated around the t′ = 0, p = 0 point,corresponding to the half-filled Hubbard model with onlynearest-neighbor hopping, in which the antinodal scatter-ing must be particle-hole symmetric by symmetry. Whenthe system is very slightly hole-doped away from p = 0,both DCA and the mean-field gauge theory suggest thatthe particle-hole symmetric point rapidly shifts to verynegative t′. This is in striking contrast to weak-couplingtheories in which approximate particle-hole symmetry atthe antinode would be restored at the non-interactingLifshitz transition (dashed line). We note that there arequantitative discrepancies in the location of these twolines between the numerical DCA results and the mean-field gauge theory results, which are predominantly dueto the assumptions made on the renormalization param-eters Zt and Zt′ entering the chargon dispersion and onthe Higgs field amplitude H0.

Importantly, the mean-field analysis of the SU(2)gauge theory provides a physical understanding of the

origin of the pseudogap and of the quasi-pole of theself-energy as being due to short-range antiferromagneticcorrelations, long-range order being destroyed by orien-tational fluctuations. The quasi-pole is responsible forthe pseudogap in the physical electron Green’s function,while the spinon (R) spectrum displays a gap. The char-gons have a spectrum characteristic of an AF spin-densitywave despite the absence of AF long-range order, andtheir self-energy has a sharp pole at mean-field level. The(red) line where the pole crosses zero energy, correspond-ing to an approximate restoration of particle-hole sym-metry at low-energy, can be interpreted39 as the Lifshitztransition of the chargon Luttinger surface.

IV. DISCUSSION AND EXPERIMENTALRELEVANCE

Our results establish that an asymmetric pole-like fea-ture in the antinodal self-energy is responsible for boththe pseudogap and for the renormalization and topologi-cal transition of the FS. We note that, in weak-couplingapproaches such as spin-fluctuation theories (see Ap-pendix E for a detailed discussion) the self-energy be-comes very large for ω = εk+(π,π) − µ, provided thatthe antiferromagnetic correlation length ξ is large enoughand that vF /ξ < T . As a result, hot spots formon the Fermi surface, at specific k-vectors defined byεk = εk+(π,π) = µ, corresponding to the intersection ofthe antiferromagnetic Brillouin zone with the Fermi sur-face. Hence, in a weak coupling approach, the change ofsign of the bare dispersion ε(π,0) − µ = 0 controls boththe doping at which the hot spots reach the antinodeand that where the Lifshitz transition occurs. As a re-sult, the non-interacting FS transition line (blue dashedline in Fig. 2) controls at the same time the location ofthe Lifshitz transition, the symmetry of the self-energyand the suppression of spectral weight along the Fermisurface. This is in stark contrast to our strong-couplingresults where these phenomena appear at distinct loca-tions. In particular, we have demonstrated that the linein (p, t′) parameter space where particle-hole symmetry isapproximately obeyed at low energy is pushed, at strongcoupling, to very low values of p and very negative valuesof t′, see Fig. 2 and Fig. 5 where this line is displayed inred. This is crucial in explaining why interactions drivethe Fermi surface more hole-like for a wide range of (p, t′)where the non-interacting (or weak coupling) FS wouldactually be electron-like, and why the Lifshitz transitionis pushed to larger values of p in comparison to the non-interacting system.

In order to put our results in perspective, we note thatthe relation between a pole-like feature in the self-energyand the pseudogap, as well as the implications of thecorresponding zeros of the Green’s function for the re-construction of the Fermi surface have been previouslydiscussed in cluster extensions of dynamical mean-fieldtheory22,23,30,32,34–36 and in phenomenological theories

8

(a) (b)

Figure 6. Pseudogap and Fermi surface topology within the SU(2) gauge theory at mean-field level. Colorcoding is identical to Fig. 2 (DCA results), to which this figure should be compared. (a) Solid blue line: Lifshitz transition ofthe interacting Fermi surface. Along the red line, the self-energy pole is at zero energy and approximate particle-hole symmetryis restored. This also corresponds to the Lifshitz transition of the chargons. (b) Electronic self-energy at the antinode for twodifferent values of t′. The quasi-pole in the self-energy moves from negative to positive frequency as t′ is increased, see Fig. 4.The inset displays the antinodal spectral function for t′ = −0.15, emphasizing that its minimum coincides with the positionof the pole. Here we assumed H0 = 0.3, a spinon gap ∆ = 0.01, and J = 0.1 (the nearest-neighbor coupling of the spin-wavefluctuations, see Ref. 39). A broadening factor η = 0.04 is used to obtain smooth spectral functions.

such as YRZ53 or other approaches54,55 (see Ref. 56 for agauge-theory perspective on the YRZ phenomenology).The existence of a Lifshitz transition as the hole dopingis increased was also discussed in some previous clusterDMFT or DCA studies30–32. However, the role playedby the particle-hole asymmetry associated with the self-energy pole in determining the FS topology, and the sys-tematic dependence of this asymmetry on (p, t′) were notunraveled and studied, and hence the key interplay be-tween FS topology and the pseudogap was not previouslyrevealed.

We now discuss the relevance of our results to exper-iments on hole-doped cuprates. We first note that, in-deed, a pseudogap is not found when the FS is electron-like and hence that the relation p∗ ≤ pFS is apparentlyobeyed in all compounds. In the single-layer compoundLa2−xSrxCuO4 (LSCO), with a small value57 of |t′/t|, thein-plane resistivity in high magnetic fields58 suggests thatp∗ ' 0.18. Currently available ARPES experiments59–61

allow to ascertain that 0.17 < pFS . 0.20. In the Nd-LSCO compound, high-field transport62 finds p∗ ' 0.23,while ARPES63 has 0.20 < pFS < 0.24.

In another single-layer compound(Bi,Pb)2(Sr,La)2CuO6+δ (Bi2201)64–67, it is foundthat p∗ ' pFS. An ARPES experiment on the bilayerBi2212 material68 has shown that the antibonding FScrosses the antinode at pFS ' 0.22 and suggested thatit may be connected to the onset of the pseudogap.This was further confirmed in a recent electronic Ramanexperiment3,4 that found the pseudogap end-point atp∗ ' 0.22. Note that the Raman response is believed tobe predominantly sensitive to the antibonding band sinceit is close to a density of states singularity3 and doesnot give information about the possible existence of apseudogap in the bonding band (which remains hole-likefor all dopings). In compounds with larger values57 of|t′/t|, such as YBa2Cu3O7−δ

69,70, Tl2Ba2CuO6+δ71,72 or

HgBa2CuO4+δ73, it is generally believed that pFS and

p∗ are distinct with p∗ < pFS. This is in qualitativeagreement with our finding that the FS and pseudogapcritical doping coincide for smaller values of |t′/t|, andare distinct for larger ones. Hence, we conclude on thebasis of our results and experimental observations thatthere are two families of hole-doped cuprates: materialswith smaller values of |t′/t| for which the collapse ofthe pseudogap and change of FS topology coincide(p∗ ' pFS), and materials with larger values of |t′/t| forwhich these are distinct phenomena (p∗ < pFS).

Finally, a very recent study on Nd-LSCO using hydro-static pressure to tune the band structure finds that bothpFS and p∗ decrease by the same amount.5 This providesa compelling experimental demonstration that p∗ cannotexceed pFS.

We finally comment on the predicted renormalizationof the FS by strong correlations. In view of Fig. 2, thematerials for which this effect is expected to be strongestare the ones with smaller values of |t′/t|, hence we turnto LSCO. We note that, in order to fit the ARPES FSusing a single-band tight binding model, the effective pa-rameter t′ has to be tuned systematically more negative(corresponding to a more negative ε(π,0)) as doping is re-duced, i.e., from t′/t = −0.12 for p = 0.3 to t′/t = −0.2for p = 0.0360. Moreover, electronic structure calcula-tions based on DFT-LDA yield pFS ' 0.15 while ARPESfinds 0.17 < pFS . 0.20, as mentioned above. Thesetwo observations suggest that correlation effects indeedgenerally drive the FS more hole-like.

V. CONCLUSION AND OUTLOOK

To conclude, we have investigated the interplay be-tween the pseudogap and the Fermi surface topologyin the two-dimensional Hubbard model. In the weak-

9

coupling regime these issues are directly connected: hot-spots can only form when the FS is hole-like and inter-sects the antiferromagnetic zone boundary. At strongercoupling, the antiferromagnetic correlations responsiblefor the pseudogap become short-ranged, and it becomesa fundamental puzzle to understand whether there isany connection to FS topology, which is in essence long-distance physics. We provide an answer to this puzzle byshowing that a common pole-like feature in the electronicself-energy controls both issues. This pole induces a largelow-energy scattering rate responsible for the onset of thepseudogap, and its asymmetry leads to significant mod-ifications of the Fermi surface with respect to its non-interacting shape and controls the location of the Lifshitztransition. As a consequence, we find that the pseudogaponly appears on hole-like Fermi surfaces, i.e. p∗ ≤ pFSand that p∗ ' pFS for an extended range of doping levelsand values of t′. These findings are in good agreementwith available experimental data. We have also shownthat our results can be interpreted in the framework ofan SU(2) gauge theory of fluctuating antiferromagnetismwith topological order. This provides an explanation forthe origin of the pole in the self-energy and establishes theconnection between the pseudogap and the Fermi surfacetopology through the chargon Luttinger surface. This ef-fort to bridge the gap between numerical results obtainedwithin cluster extensions of DMFT and low-energy effec-tive field theories is pursued and detailed in a companionpublication39.

Let us emphasize that in most of the parameter rangerelevant to hole-doped cuprates, the self-energy pole isfound at a positive energy. Hence, the strongest sup-pression of the antinodal spectral weight is predicted tooccur at energies above the Fermi level, which is not di-rectly accessible to ARPES experiments. While a strongparticle-hole asymmetry is indeed observed by STM74,75,this emphasizes again76 the importance of developingmomentum-resolved spectroscopies able to probe the‘dark side’ of the FS. Finally, an outstanding questionis to explore whether the topological order, associatedwith the pseudogap regime in the gauge theory descrip-tion, can be revealed more directly in numerical studiesof Hubbard-like models.

ACKNOWLEDGMENTS

We are grateful to L. Taillefer and N. Doiron-Leyraudfor sharing and discussing their experimental data be-fore publication. We also acknowledge discussions withM. Civelli, M. Kim, G. Kotliar, A. J. Millis, O. Par-collet, I. Paul, A. Sacuto and A.-M.S. Tremblay. Thiswork was supported by the Simons Foundation Many-Electron Collaboration, the European Research Council(project ERC-319286-‘QMAC’), the Swiss National Su-percomputing Centre (CSCS, project s575), the NSF un-der Grant DMR-1664842 and MURI grant W911NF-14-1-0003 from ARO. The Flatiron Institute is supported by

the Simons Foundation (AG). Research at Perimeter In-stitute (SS) is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation. MS ac-knowledges support from the German National Academyof Sciences Leopoldina through grant LPDS 2016-12. SSacknowledges support from Cenovus Energy at Perime-ter Institute, and from the Hanna Visiting Professor pro-gram at Stanford University. AG and SS are grateful forthe stimulating atmosphere of the May 2017 Jouvenceworkshop.

Appendix A: Methods

Our results for the two-dimensional Hubbard modelare obtained using two methods: unbiased determinantquantum Monte Carlo (DQMC40) and the dynamicalcluster approximation (DCA15,77), a cluster extensionof dynamical mean-field theory (DMFT78) that capturesthe physics of short-range spatial correlations. We per-form DQMC on a 16× 16 lattice with periodic boundaryconditions at a temperature T = 1/3. Since the inversetemperature β = T−1 = 3 is significantly smaller thanthe linear size of the lattice L = 16, the finite size effectsare negligible in the DQMC calculation. The imaginarytime step was set to ∆τ = 3/64 which is small enough toavoid artifacts due to the discretization errors. We use5.12 × 105 Monte Carlo sweeps to collect the data after1000 warmup sweeps.

The DCA calculation is performed with an eight-sitecluster. In the DCA approach, the lattice self-energy isapproximated by a patchwise-constant self-energy ΣK inthe Brillouin zone. We solved the DCA equations withan eight-site auxiliary quantum impurity cluster. In thegeometry we used, the Brillouin zone is divided in eightsectors where the self-energy is constant, as shown inFig. 7. Note that there are clearly distinct patches forthe antinodal and the nodal region of the Brillouin zone.

We use both the the Hirsch-Fye77 and the continuous-time quantum Monte Carlo79 method to solve the aux-iliary cluster impurity problem. A comparison of bothmethods shows that the imaginary-time step ∆τ = 1/21

(0,0)(π,0)

(π,π)(0,π)

Figure 7. The patches in momentum space of the eight-siteDCA method. The self-energy is constant over each patch.

1 0

u s e d i n t h e Hi r s c h- F y e s ol v er i s s m all e n o u g h, a s s h o w ni n t h e A p p e n di x B. We u s e 5 0 D M F T it er ati o n s t o g etc o n v er g e d r e s ult a n d u s e ( 2 ∼ 1 0) × 1 0 6 M o nt e C a rl os w e e p s at e a c h it er ati o n. I n or d er t o h a v e b ett er st ati s-ti c s, t h e r e s ult s ar e a v e r a g e d o v er t h e l a st f e w c o n v er g e dit er ati o n s. T h e t y pi c al st ati sti c al er r or i n t h e r e al p artof t h e s elf- e n er g y a n d t h e s p e ct r al i nt e n sit y at z er o f r e-q u e n c y i s ∼ 1 %.

We i d e ntif y t h e p s e u d o g a p t e m p er at u r e T ∗ a s t h em a xi m u m of t h e t e m p er at u r e d e p e n d e nt s p e ct r al w ei g htA ( π, 0 ) ( ω = 0) ≡ − 1

π I mG ( π, 0 ) ( ω = 0). It i s o b-t ai n e d f r o m a li n e ar e xt r a p ol ati o n t o z er o f r e q u e n c y of− I mG ( π, 0 ) ( i ωn ) at t h e fi r st t w o M at s u b ar a f r e q u e n ci e s.We fi n d T F S f r o m t h e z e r o of t h e e ff e cti v e di s p er si o n att h e a nti n o d e ˜( π, 0 ) .

Fi n all y, t h e r e al-f r e q u e n c y s p e ct r al f u n cti o n A k ( ω ) a n dt h e s elf- e n er g y Σ k ( ω ) a r e f o u n d wit h t h e m a xi m u m e n-t r o p y a n al y si s 8 0, 8 1 o n t h e G r e e n’ s f u n cti o n G k ( i ωn ) a n dt h e s elf- e n er g y Σ k ( i ωn ). We u s e t w o i n d e p e n d e nt m a x-i m u m e nt r o p y c o d e s8 0, 8 1 t o m a k e s u r e t h at t h ei r r e s ult sa gr e e.

A p p e n di x B: S u p p o r ti n g m a t e ri al f o r t h e r e s ul t ss h o w n i n t h e m ai n t e x t

1. P s e u d o g a p o n s e t t e m p e r a t u r e T ∗ a n d F e r mis u rf a c e t o p ol o g y t r a n si ti o n t e m p e r a t u r e T F S

We i d e ntif y t h e p s e u d o g a p o n s et t e m p er at u r e T ∗ a s t h et e m p e r at u r e w h er e t h e s p e ct r al f u n cti o n at t h e a nti n o d alp oi nt A ( π, 0 ) ( ω = 0) r e a c h e s a m a xi m u m a s t e m p er at u r ei s l o w er e d, s e e Fi g. 8. T h e z er o-f r e q u e n c y v al u e of t h es p e ct r al f u n cti o n A ( π, 0 ) ( ω = 0) = − 1

π I mG ( π, 0 ) ( ω = 0)w hi c h i s o bt ai n e d b y a li n e ar e xt r a p ol ati o n of t h e v al u eof I m G ( π, 0 ) ( i ωn ) (t h e r e s ult of t h e n u m eri c al c al c ul ati o n)at it s fi r st t w o M at s u b ar a fr e q u e n ci e s. We f o u n d t h atdi ff er e nt a p p r o xi m ati o n s of t h e s p e ct r al f u n cti o n, s u c ha s A ( π, 0 ) ( ω = 0) β G ( π, 0 ) ( τ = β

2 ) 8 2 yi el d t h e s a m ev al u e s of T ∗ . Al s o, w e h a v e u s e d di ff e r e nt i m a gi n ar yti m e di s cr eti z ati o n st e p s ∆ τ i n t h e Hi r s c h- F y e al g orit h ma n d o b s er v e t h at t h e r e s ult s ar e t h e s a m e f or all v al u e sof ∆ τ ≤ 0 .1. We h a v e u s e d ∆ τ = 0 .0 4 7 6 t h r o u g h o ut o u rw or k.

W h e n t h e Fer mi s u rf a c e cr o s s e s k = ( π, 0) it u n d er-g o e s a Lif s hit z t r a n siti o n a n d c h a n g e s f r o m h ol e-li k e t oel e ct r o n-li k e. We d e fi n e t h e Fer mi s u rf a c e at fi nit e t e m-p er at u r e b y t h e l o c ati o n of t h e m a xi m u m s p e ct r al i nt e n-sit y at z er o e n er g y ( a s s e e n i n a n A R P E S e x p eri m e nt).T hi s m a xi m u m g o e s t h r o u g h ( π, 0) w h e n t h e q u a si p ar-ti cl e e ff e cti v e di s p er si o n at t h e a nti n o d e ˜( π, 0 ) = 0 a sdi s c u s s e d i n t h e m ai n t e xt ( s e e E q u ati o n 1). We c a nt h er ef or e fi n d t h e Fer mi s u rf a c e t o p ol o g y t r a n siti o n t e m-p er at ur e T F S b y fi n di n g t h e z er o of ˜ ( π, 0 ) a s a f u n cti o nof t e m p er at u r e a s s h o w n i n Fi g. 9. We al s o di s pl a y t h er e s ult s a s o bt ai n e d u si n g t w o di ff er e nt i m p u rit y s ol v er sa n d s h o w t h at t h e y gi v e i d e nti c al r e s ult s.

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0

T

0 .5

1 .0

−I

mG (

π ,0)(ω

= 0)

T ∗

T ∗

T ∗

t = − 0 .1 , U = 7p = 0 .0 8

p = 0 .1 0

p = 0 .1 2

p = 0 .1 5

0 .0 0 0 .0 5 0 .1 0

T

0 .5

0 .6

0 .7

−I

mG(π

,0)

(ω=0

)

t = − 0 .2 , p = 0 .1 , U = 7

T ∗

∆ τ = 0 .0 4 7 6

∆ τ = 0 .0 7 1 4

∆ τ = 0 .1

Fi g u r e 8. − I mG ( π , 0 ) ( ω = 0 ) a s a f u n c ti o n of t e m p e r a t u r e T .T h e m a xi m u m i s i d e nti fi e d a s t h e p s e u d o g a p o n s e t t e m p e r a-t u r e T ∗ . U p p e r p a n el: f o r t = − 0 .1 a n d di ff e r e nt d o pi n gl e v el s. L o w e r p a n el: f o r t = − 0 .2 , p = 0 .1 a n d di ff e r e ntdi s c r e t e ti m e s t e p s ∆ τ of t h e Hi r s c h- F y e i m p u ri t y s ol v e r. Wec a n s e e t h a t T ∗ i s al r e a d y c o n v e r g e d f o r ∆τ = 0 .1.

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0

T

− 0 .1 5

− 0 .1 0

− 0 .0 5

0 .0 0

0 .0 5

0 .1 0

˜ (π

,0)

p = 0 .1 5

p = 0 .1 6T F S T F S

Hi r s c h- F y e

C o nti n u o u s ti m e

Fi g u r e 9. T h e e ff e c ti v e q u a si p a r ti cl e di s p e r si o n a t t h e a nti n-o d e ˜ ( π , 0 ) a s a f u n c ti o n of t e m p e r a t u r e T f o r t w o d o pi n g l e v el s.At t h e Fe r mi s u rf a c e t o p ol o g y t r a n si ti o n t e m p e r a t u r e T F S , w eh a v e ˜ ( π , 0 ) = 0. We s h o w r e s ul t s u si n g b o t h a c o nti n u o u s- ti m e

Q M C 7 9 a n d a Hi r s c h- F y e 7 7 i m p u ri t y s ol v e r ( t h e l a t t e r wi t h afi ni t e i m a gi n a r y- ti m e s t e p ∆ τ = 0 .0 4 7 6 ).

1 1

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5

T

− 6

− 4

− 2

0

2

4

6

8

ReΣ

(2)

(π

,0)

U = 7 , t = − 0 .1

p = 0 .0 1

p = 0

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5

T

− 1 .0

− 0 .5

0 .0

0 .5

1 .0

1 .5

ReΣ

(2)

(π

,0)

U = 7 , t = − 0 .2

p = 0 .0 6

p = 0 .0 4

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5

T

− 4

− 2

0

2

4

6

8

ReΣ

(2)

(π

,0)

U = 7 .5 , t = − 0 .2p = − 0 .0 1

p = 0 .0 1

Fi g u r e 1 0. T h e r e al p a r t of t h e a nti n o d al s elf- e n e r g y a s a f u n c ti o n of t e m p e r a t u r e f o r di ff e r e nt d o pi n g s p a n d t . N o t e t h a t

t h e c o n s t a nt H a r t r e e s hif t h a s b e e n r e m o v e d. L ef t p a n el : F o r U = 7 , t = − 0 .1, a t p = 0, R e Σ( 2 )

( π , 0 ) ( ω = 0 ) i s p o si ti v e a t l o w

t e m p e r a t u r e s w hil e a t p = 0 .0 1 i t i s n e g a ti v e, s u g g e s ti n g p c i s b e t w e e n 0 a n d 0. 0 1. C e n t r al p a n el : F o r U = 7 , t = − 0 .2, w eh a v e 0 .0 4 < p c < 0 .0 6. Ri g h t p a n el : F o r U = 7 .5 , t = − 0 .2, w e fi n d − 0 .0 1 < p c < 0 .0 1. T h e r ef o r e f o r fi x e d t , i n c r e a si n g Ud ri v e s p c cl o s e r t o t h e 0 ( h alf- filli n g ).

2. M a xi m u m l o w - e n e r g y s c a t t e ri n g li n e a n dp a r ti cl e - h ol e a s y m m e t r y

A s w e h a v e s h o w n i n t h e m ai n t e xt, t h e r e i s a c u r v e i nt h e p − t di a gr a m t h at s e p ar at e s a r e gi o n w h er e t h e p ol e-li k e f e at u r e i n t h e i m a gi n ar y p art of t h e a nti n o d al s elf-e n er g y i s o n t h e n e g ati v e e n er g y si d e f r o m a r e gi o n w h er eit i s o n t h e p o siti v e e n er g y si d e. W h e n t h e p ol e i s o n t h ep o siti v e e n er g y si d e, t h e r e al p art of t h e s elf- e n er g y ( wit h

t h e H art r e e t er m r e m o v e d) R e Σ( 2 )( π, 0 ) ( ω = 0) i s n e g ati v e,

w hil e it i s p o siti v e w h e n t h e p ol e i s o n t h e n e g ati v e si d e.We c a n t h er ef or e l o c at e t h e c u r v e b y fi n di n g, at fi x e d t ,t h e v al u e of t h e d o pi n g p c a t w hi c h t h e z e r o-t e m p er at u r e

e xt r a p ol ati o n of R e Σ( 2 )( π, 0 ) ( ω = 0) c h a n g e s si g n s e e Fi g. 1 0.

3. U d e p e n d e n c e of t h e c o n n e c ti o n b e t w e e n p ∗ a n dp F S

I n t h e m ai n t e xt, all c al c ul ati o n s h a v e u s e d U = 7.F or t hi s v al u e, w e h a v e s h o w n t h at, p ∗ p F S f o r v al u e sof t g r e at er t h a n − 0 .1. F or m or e n e g ati v e v al u e s oft t h e p ∗ a n d p F S li n e s s plit a p art. W h e n U i s l ar g er,t hi s b r a n c hi n g p oi nt g o e s t o l o w er v al u e s of t . T hi s i ss h o w n i n Fi g. 1 1 w h er e w e c o m p ut e T ∗ a n d T F S f o r b ot hU = 7 a n d U = 7 .5. It i s cl e ar f r o m t h e fi g u r e, t h at f orU = 7 .5 p ∗ a n d p F S a r e m u c h cl o s er t h a n f or U = 7. T hi sc a n b e u n d er st o o d b e c a u s e a l ar g er v al u e of U e xt e n d st h e p s e u d o g a p r e gi o n t o l ar g er d o pi n g s, w hil e t h e Fe r mis u rf a c e t o p ol o g y i s n ot i n fl u e n c e d m u c h b y c or r el ati o n sb ef or e w e a ct u all y h a v e a p s e u d o g a p a n d h e n c e p ∗ p F S .

A p p e n di x C: T e m p e r a t u r e e v ol u ti o n of ˜ k a n d r ol e ofn o n -l o c al c o r r el a ti o n s

I n Fi g. 1 2, w e i n v e sti g at e t h e r ol e of n o n-l o c al c or-r el ati o n s b y c o m p ari n g t h e r e s ult s o bt ai n e d b y D C A,a s i n t h e m ai n t e xt, a n d si n gl e- sit e d y n a mi c al m e a n-fi el d ( D M F T) t h at o nl y a c c o u nt s f or l o c al c or r el ati o n s.

0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0p

0 .0

0 .1

0 .2

0 .3

T

U = 7 .0

U = 7 .5

t = − 0 .2

T ∗

T F S

Fi g u r e 1 1. T ∗ a n d T F S a s a f u n c ti o n of d o pi n g p a t t = − 0 .2,U = 7 ( d a s h e d li n e s ), a n d U = 7 .5 ( s oli d li n e s ). N o t e t h a th e r e t h e i m a gi n a r y- ti m e di s c r e ti z a ti o n s t e p i s ∆ τ = 1 / 2 U .

T h e bl a c k li n e s h o w ˜ ( π, 0 ) a s c o m p ut e d b y D M F T f ort = − 0 .1 a n d p = 0 .1. F or t h e s e p ar a m et er s, t h e n o n-i nt er a cti n g Fer mi s u rf a c e i s el e ct r o n-li k e. It i s s e e n t h atat l o w t e m p er at u r e t h e D M F T r e s ult s al s o p r e di ct a nel e ct r o n-li k e Fer mi s u rf a c e. T hi s i s n ot s u r p ri si n g a sD M F T p r e s er v e s t h e L utti n g er t h e or e m a n d t h e i nt er-a cti n g Fer mi s u rf a c e i s t h e s a m e a s t h e n o n-i nt er a cti n go n e w h e n T → 0. H o w e v er, a s t e m p er at u r e i s i n cr e a s e d,˜ ( π, 0 ) d e c r e a s e s si g ni fi c a ntl y a n d b e c o m e s n e g ati v e. T hi syi el d s a h ol e-li k e i nt er a cti n g Fer mi s urf a c e at hi g h t e m-p er at ur e t h at b r e a k s L utti n g er’ s t h e or e m 8 3, 8 4 wi t h a v ol-u m e l ar g er t h a n i n t h e n o n-i nt er a cti n g c a s e.

T h e r e d a n d bl u e li n e s s h o w ˜ ( π, 0 ) a n d ˜ ( π2 , π

2 ) r e s p e c-ti v el y a s o bt ai n e d b y D C A. ˜( π

2 , π2 ) h a s b e e n s hift e d b y

a c o n st a nt 4 t = − 0 .4 t h at c or r e s p o n d s t o t h e e n e r g ydi ff er e n c e of t h e n o n-i nt er a cti n g di s p er si o n at ( π, 0) a n d( π

2 , π2 ). At hi g h t e m p er at u r e s all c u r v e s yi el d t h e s a m e

v al u e, c o m p ati bl e wit h a s elf- e n e r g y t h at i s e s s e nti all yl o c al. A s t e m p er at u r e i s d e cr e a s e d t h e n o d al ( π

2 , π2 ) b e-

h a v e s li k e t h e D M F T s ol uti o n i n di c ati n g t h at t h e Fer mi

1 2

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2

T

− 1

0

t = − 0 .1 , p = 0 .1 , U = 7

T ∗

D M F T, ˜(π ,0)

D C A, ˜(π ,0)

D C A, ˜( π2 ,π

2 ) - 0. 4

Fi g u r e 1 2. Te m p e r a t u r e d e p e n d e n c e of ˜ k a s o b t ai n e d b yD C A a n d D M F T a t 1 0 % h ol e d o pi n g a n d t = − 0 .1.

s u rf a c e at t h e n o d e i s v er y cl o s e t o it s n o n-i nt er a cti n gs h a p e. T h e D C A ˜( π, 0 ) h a s a di ff e r e nt b e h a vi or. A s t e m-p er at u r e i s l o w e r e d it q ui c kl y d e p art s f r o m ˜ ( π

2 , π2 ) s h o w-

i n g t h e o n s et of n o d al / a nti n o d al di ff er e nti ati o n. At at e m p er at u r e sli g htl y a b o v e T ∗ , n o n-l o c al c or r el ati o n s b e-c o m e l ar g e a n d i n d u c e a v er y n e g ati v e ˜ ( π, 0 ) a s di s c u s s e di n t h e m ai n t e xt.

A p p e n di x D: P s e u d o g a p a n d F e r mi s u rf a c e t o p ol o g yt r a n si ti o n a s a f u n c ti o n of U

O u r r e s ult s f or U = 7 s h o w t h at, f or a b r o a d r a n g e ofp ar a m et er s, t h e p s e u d o g a p di s a p p e ar s at t h e s a m e c riti-c al d o pi n g w h e r e t h e Fer mi s u rf a c e u n d er g o e s a Lif s hit zt r a n siti o n. I n Fi g. 1 3, w e i n v e sti g at e h o w o ur r e s ult sd e p e n d o n t h e c or r el ati o n st r e n gt h U . It i s s h o w n t h atf or v al u e s of U 5 c or r el ati o n s h a v e littl e e ff e ct o n t h eFer mi s u rf a c e t o p ol o g y a n d t h e l o w- e n e r g y s c att eri n g r at eI m Σ( π, 0 ) i s v e r y s m all. A b o v e U 5 c or r el ati o n e ff e ct ss et i n q ui c kl y a s s h o w n b y a f a st i n cr e a s e i n t h e v al u eof I m Σ ( π, 0 ) . T hi s i n d u c e s a p s e u d o g a p at U = 5 .6. Att h e s a m e ti m e t h e e ff e cti v e q u a si p arti cl e di s p er si o n ˜( π, 0 )

c r o s s e s z e r o a n d b e c o m e s v er y n e g ati v e f or l ar g er v al u e sof U . T hi s s u d d e n i n cr e a s e of t h e c or r el ati o n e ff e ct s f orU > 5 mi g ht e x pl ai n w h y t h e p s e u d o g a p a n d t h e Fer mis u rf a c e t o p ol o g y h a p p e n at t h e s a m e ti m e.

A p p e n di x E: C o m p a ri s o n wi t h w e a k - c o u pli n ga p p r o a c h e s

L et u s i n v e sti g at e h o w o u r r e s ult s di ff er f r o m w e a k-c o u pli n g a p p r o a c h e s, s u c h a s s pi n- fl u ct u ati o n t h e or y ort h e t w o- p arti cl e s elf- c o n si st e nt a p p r o a c h ( T P S C) of Vil ka n d Tr e m bl a y 8 5 . T h e l att e r h a s b e e n s h o w n t o b e q uit ea c c u r at e i n t h e w e a k t o i nt er m e di at e c o u pli n g r e gi m e oft h e t w o- di m e n si o n al H u b b ar d m o d el, a n d t hi s a p p e n di x

2 3 4 5 6 7

U

− 0 .5

0 .0

0 .5t = 0 , p = 0 .0 5

T = 0 .0 2 5

El e ct r o n-li k e F S

H ol e-li k e F SWit h o ut P s e u d o g a p

Wit h P s e u d o g a p

∆ ˜A (π ,0) (ω = 0) |T 2 = 0 .0 3 3T 1 = 0 .0 2 5

˜ (π ,0)

I m Σ (π ,0)

Fi g u r e 1 3. C o r r el a ti o n e ff e c t s s h o w n a s a f u n c ti o n of U f o rt h r e e q u a nti ti e s: t h e q u a si p a r ti cl e e ff e c ti v e di s p e r si o n ˜( π , 0 ) ,t h e i m a gi n a r y p a r t of t h e a nti n o d al s elf- e n e r g y a t z e r o e n e r g yI m Σ( π , 0 ) ( ω = 0 ) a n d t h e di ff e r e n c e i n s p e c t r al i nt e n si t y a tt h e Fe r mi l e v el f o r t h e t w o l o w e s t c al c ul a t e d t e m p e r a t u r e si n di c a ti n g w h e t h e r a p s e u d o g a p h a s f o r m e d.

cl o s el y f oll o w s t h e a n al y si s i n R ef. 8 5.I n t h o s e a p p r o a c h e s, t h e s elf- e n er g y i s o bt ai n e d a s

Σ( k , i ωn ) = g 2 Tp

1

V B Z q

G 0 ( k + q , i ωn + i νp ) χ (q , i νp ) ,

( E 1)w h e r e G 0 i s t h e n o n-i nt er a cti n g G r e e n’ s f u n cti o n, χ i st h e s pi n s u s c e pti bilit y, a n d g i s a c o u pli n g c o n st a nt wit ht h e di m e n si o n of e n er g y. W h e n t h e m a g n eti c c or r el ati o nl e n gt h ξ i s l ar g e, χ c a n b e a p p r o xi m at e d b y: 8 6

χ ( q , i νp ) ∝1

(q − Q ) 2 + ξ − 2 ν p / ω s p + ξ − 2, ( E 2)

wit h Q = ( π, π ) t h e a ntif er r o m a g n eti c w a v e- v e ct or. T h es elf- e n er g y t h u s r e a d s

Σ( k , i ωn ) ∝ Tp

1

V B Z q

1

i ωn + i νp + µ − k + q

×1

( q − Q ) 2 + ξ − 2 ν p / ω s p + ξ − 2.

( E 3)

I n t h e r e gi m e of i nt er e st h er e (l ar g e e n o u g h ξ , r e n or m al-i z e d cl a s si c al r e gi m e), t h e a b o v e s u m i s d o mi n at e d b yt h e s m all e st M at s u b ar a f r e q u e n c y ( n ot e t h at ω s p ∼ ξ − 2 )a n d o n e o bt ai n s t h e i m a gi n ar y p art of t h e r et ar d e d r e al-f r e q u e n c y s elf- e n er g y i n t h e f or m:

−1

πI m Σr e t ( k , ω) ∝ T d 2 q δ (ω − ξ k + q )

1

( q − Q ) 2 + ξ − 2

( E 4)T h e i m p o rt a nt p oi nt i s t h at i n t w o di m e n si o n s, t hi s i n-t e gr al di v er g e s a s t h e c or r el ati o n l e n gt h b e c o m e s l ar g e,w hi c h l e a d s t o t h e f or m ati o n of ‘ h ot s p ot s’ at w hi c h a

1 3

p s e u d o g a p o p e n s. T hi s i nt e gr al c a n a ct u all y b e p er-f or m e d a n al yti c all y, a n d o n e fi n all y o bt ai n s:

−1

πI m Σr e t ( k , ω) = g

T

( ω − ξ k + Q ) 2 + ( v F / ξ ) 2+ r e g .

( E 5)w h e r e ‘ r e g.’ d e n ot e s a n o n- si n g ul ar c o nt ri b uti o n. T h ep h y si c s a s s o ci at e d wit h a w e a k- c o u pli n g d e s cri pti o n ofs pi n fl u ct u ati o n s c a n b e e ntir el y d e s cri b e d o n t h e b a si sof t hi s e x p r e s si o n 8 5 . L et u s f o c u s fi r st o n t h e Fer mi s u r-f a c e pr o p erti e s, c orr e s p o n di n g t o ω = 0 a n d m o m e nt as u c h t h at ξ k = 0. A s cl e a r f r o m ( E 5), t h e s elf- e n er g y i sr e g ul ar o n t h e Fer mi s u rf a c e e x c e pt at t h e ‘ h ot s p ot s’ s at-i sf yi n g al s o ξ k + Q = 0, c o r r e s p o n di n g t o t h e i nt er s e cti o nof t h e Fer mi s urf a c e wit h t h e a ntif er r o m a g n eti c Brill o ui nz o n e. At t h e s e h ot- s p ot s, t h e s elf- e n er g y i s si n g ul ar: it si m a gi n ar y p art i s of or d er:

−1

πI m Σ|h o t ∝

T ξ

v F( E 6)

T hi s i s l ar g e o nl y w h e n t h e c or r el ati o n l e n gt h i s l ar g e:ξ > v F / T . I n t hi s r e gi m e, s p e ct r al w ei g ht i s st r o n gl yd e pl et e d at t h e h ot s p ot s, c or r e s p o n di n g t o t h e w e a k-c o u pli n g d e s c ri pti o n of t h e p s e u d o g a p. W h e n t h e c or-r el ati o n l e n gt h r e m ai n s fi nit e a s t e m p er at u r e i s l o w e r e d( s h ort-r a n g e or d er), t h e h ot s p ot s a n d c or r e s p o n di n gp s e u d o g a p di s a p p e ar f or T < v F / ξ a n d c o n v e nti o n alFer mi li q ui d b e h a vi or i s r e c o v er e d at l o w t e m p er at ur e.

L et u s e m p h a si z e t h e cr u ci al di ff er e n c e s t h at e xi st b e-t w e e n t h e w e a k- c o u pli n g e x p r e s si o n of t h e s elf- e n er g y( E 5) a n d b ot h t h e s elf- e n er g y t h at w e o bt ai n f r o m D C Aat st r o n g c o u pli n g, a s w ell a s t h e s elf- e n er g y o bt ai n e df r o m t h e S U( 2) g a u g e t h e or y. A s cl e ar fr o m ( E 5), t h ei m a gi n ar y p art of t h e w e a k- c o u pli n g s elf- e n er g y d o e s di s-pl a y a p e a k, b ut (i) t h e h ei g ht of t hi s p e a k i s pr o p or-ti o n al t o T ξ (T ) a n d t h u s e v e nt u all y t h e p e a k a n d t h eh ot s p ot s di s a p p e ar at l o w- T if ξ r e m ai n s fi nit e (ii) t h ewi dt h of t hi s p e a k i s p r o p orti o n al t o v F / ξ < T , w hi c hi n t h e r e gi m e w h er e t h e p e a k e xi st s i s s m all er t h a n t e m-p er at ur e. I n c o nt r a st, i n t h e st r o n g- c o u pli n g D C A c al-c ul ati o n s t h e p e a k i s n ot s u p p r e s s e d a s T i s r e d u c e d,a n d it s wi dt h i s l ar g er t h a n T . F u rt h er m or e, t h e c or-r el ati o n l e n gt h t h at w e c a n e sti m at e i n o ur D C A r e s ult sf r o m t h e st ati c st a g g er e d s u s c e pti bilit y χ A F ∝ ξ 2 i s q uit es m all at st r o n g c o u pli n g: w e fi n d f or e x a m pl e: ξ / a 2 .7f or U = 7 , t = − 0 .2 , p = 0 .1 at T = 1 / 3 0. T h e w e a k-c o u pli n g e x p r e s si o n al s o h a s a di ff er e nt st r u ct u r e t h a n t h esi n g ul ar d elt a-f u n cti o n f or m of t h e c h ar g o n s elf- e n er g y i nt h e S U( 2) g a u g e t h e or y: t h e l att er, i m p ort a ntl y, d o e s n oti n v ol v e t h e c or r el ati o n l e n gt h ( s et b y t h e s pi n o n s) a n d i ssi mil ar t o t h at of a n S D W i n t h e or d er e d p h a s e.

T h e r e al p art of t h e s elf- e n er g y c or r e s p o n di n g t o ( E 5)c a n b e o bt ai n e d u si n g K r a m er s- K r o ni g r el ati o n s a s:

R e Σ r e t ( k , ω) ∝T

Ω 2k + ( v F / ξ ) 2

× l n |Ω k + Ω 2

k + ( v F / ξ ) 2

Ω k − Ω 2k + ( v F / ξ ) 2

|

( E 7)

0 .0 0 .1 0 .2

p

− 0 .3

− 0 .2

− 0 .1

0 .0

0 .1

0 .2

t

1 6 × 1 6 D Q M C, T = 1/ 3

U = 4

U = 6

U = 8

(π , π ) S D W

Fi g u r e 1 4. T h e li n e s i n t h e t − p pl a n e s h o w w h e r et h e l o w- e n e r g y i m a gi n a r y p a r t of t h e a nti n o d al s elf- e n e r g yI m Σ( π , 0 ) ( ω ) h a s i t s p ol e-li k e f e a t u r e c e nt e r e d a r o u n d ω = 0a n d a n e s s e nti all y p a r ti cl e- h ol e s y m m e t ri c l o w- e n e r g y s p e c-t r u m. O n t h e s e c u r v e s R e Σ ( π , 0 ) ( ω = 0 ) v a ni s h e s. T h e s oli dli n e s a r e o b t ai n e d b y D Q M C f o r di ff e r e nt v al u e s of U , w hil et h e d a s h e d li n e i s t h e r e s ul t f r o m t h e S D W w e a k- c o u pli n g a p-p r o a c h. T h e d a s h e d li n e c oi n ci d e s wi t h t h e n o n-i nt e r a c ti n gLif s hi t z t r a n si ti o n.

i n w hi c h w e h a v e u s e d t h e s h ort- h a n d n ot ati o n Ωk ≡ω − ξ k + Q . I n t h e t e m p er at u r e r e gi m e w h er e h ot s p ot sar e p r e s e nt T > v F / ξ , o n e c a n di sti n g ui s h t w o r e gi m e sof f r e q u e n ci e s. F or Ω k > v F / ξ , R e Σ i s of or d er(T / Ω k ) l n( Ω k ξ / v F ), w hil e at l o w f r e q u e n ci e s Ω k < v F / ξ ,t h e s elf- e n e r g y i s r e g ul ar R e Σ ∝ T Ω k ( ξ / v F ) 2 . H e n c e, t h er e g ul ar p art of t h e s elf- e n er g y i s r e g ul ar at l o w-f r e q u e n c ye v e n cl o s e t o t h e h ot s p ot s w h e n ξ r e m ai n s fi nit e. A sa r e s ult, h ot s p ot s e xi st at t h e i nt er s e cti o n b et w e e n t h eFer mi s u rf a c e a n d t h e a ntif er r o m a g n eti c Brill o ui n z o n ef or T > v F / ξ , b ut t h e r e i s n o r e c o n st r u cti o n of t h e Fer mis u rf a c e ot h er wi s e. H e n c e, f or a d o pi n g v al u e l ar g er t h a nt h e v al u e c or r e s p o n di n g t o t h e n o n-i nt er a cti n g Lif s hit zt r a n siti o n, t h er e ar e n o h ot s p ot s a n d t h e Fer mi s u rf a c ei s w e a kl y r e n or m ali z e d a n d el e ct r o n-li k e. H e n c e, i n w e a k-c o u pli n g t h e n o n-i nt er a cti n g Lif s hit z t r a n siti o n c o nt r ol sb ot h t h e l o c ati o n of t h e s elf- e n er g y si n g ul ariti e s a n d t h et o p ol o gi c al t r a n siti o n of t h e Fer mi s u rf a c e. T hi s i s v er ydi ff er e nt f r o m o u r r e s ult s i n t h e st r o n g c o u pli n g r e gi m eU = 7 w h er e t h e s e p h e n o m e n a ar e c o nt r oll e d b y t h r e edi ff er e nt li n e s.

B y v ar yi n g U o n e c a n o b s er v e h o w t h e t r a n siti o n f r o mw e a k t o st r o n g c o u pli n g h a p p e n s. Fi g. 1 4 s h o w s t h eD Q M C r e s ult s f or s e v er al v al u e s of U . T h e li n e s s h o ww h er e t h e r e al p art of t h e s elf- e n er g y v a ni s h e s. It s e p a-r at e s a r e gi o n w h er e t h e p ol e i n t h e s elf- e n er g y i s at n e g-ati v e e n er gi e s a n d o n e w h er e it i s o n t h e p o siti v e si d e.It i s s e e n t h at a s U b e c o m e s s m all er t h e li n e s sl o wl ya p p r o a c h t h e n o n-i nt er a cti n g Lif s hit z t r a n siti o n, a s e x-p e ct e d i n w e a k- c o u pli n g.

14

x10

Figure 15. Scattering rate of the electrons −ImΣk(ω = 0)

and chargons −ImΣchargonk (ω = 0) in momentum space. Theparameters used here are: H0 = 0.2, J = 0.1;T = 1/30, t′ =−0.3, Zt = 0.31, Zt′ = 0.19,∆ = 0.01, p = 0.05. Left panel:The physical electron self-energy has clear nodal/antinodaldifferentiation with a stronger scattering at the antinode thanat the node. Right panel: The chargon self-energy is givenby Eq. 8 and has no momentum differentiation. The broad-ening in the chargon self-energy is η = 0.04.

Appendix F: Comparison of the chargon andelectron self-energy in the SU(2) gauge theory

Here we illustrate in more details the role of the convo-lution that allows to recover the electronic Green’s func-tion in the SU(2) theory. As we have discussed above,see e.g. Fig. 6, the location in momentum and fre-quency of the most singular structures of the physicalself-energy are not affected by the convolution and theyare already encoded in the chargon self-energy given byEq. 8. The convolution mainly smears Gψ and the elec-tron self-energy is a broadened counterpart of the char-gon self-energy. A more detailed inspection shows thatthe convolution also redistributes spectral weight overthe Brillouin zone. As a result, the physical electron self-energy displays nodal/antinodal differentiation, which isabsent in the chargon self-energy. This is illustrated inFig. 15 where it is clearly seen that the imaginary part ofthe electronic self-energy is larger close to the antinodethan at the node. This differentiation is not present inthe chargon self-energy.

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