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

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 the existence of the pseudogap and the topology of the Fermi surface. Here, we address this issue by studying the two-dimensional Hubbard model with two distinct numerical methods. We find that the pseudogap only exists when the Fermi surface is hole-like and that, for a broad range of 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 the electronic self-energy associated with the formation of the pseudogap is found to also control the degree of particle-hole asymmetry, and hence the Fermi surface topology transition. We interpret our results in the framework of an SU(2) gauge theory of fluctuating antiferromagnetism. We show that a mean-field treatment of this theory in a metallic state with U(1) topological order provides an explanation of this pole-like feature, and a good description of our numerical results. We discuss the relevance of our results to experiments on cuprates.


I. INTRODUCTION
A very debated topic in the physics of hightemperature superconductors is the nature of the 'pseudogap' 1,2 in their phase diagram. Below a temperature T * (p) which is a decreasing function of the holedoping level p, a pseudogap develops, corresponding to a suppression of low-energy excitations apparent in many experimental probes. Extrapolated to zerotemperature, T * (p) defines a critical hole doping p * above which the pseudogap disappears as doping is increased. Another important critical value of the doping, denoted here p FS , is that at which the Fermi surface topology changes from hole-like to electron-like, corresponding to a Lifshitz transition. Recent experiments on Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212) have suggested that the pseudogap may be very sensitive to the Fermi surface (FS) topology and that p * p FS in this compound 3,4 . In a simultaneous and independent manner from the present theoretical work, Doiron-Leyraud et al. 5 recently performed a systematic experimental study using hydrostatic pressure as a control parameter in the La 1.6−x Nd 0.4 Sr x CuO 4 (Nd-LSCO) system, and an unambiguous connection between FS topology and the pseudogap was found.
In this work, we investigate this interplay by studying the two-dimensional Hubbard model. In the weakcoupling scenarios of pseudogap physics, there is a natural connection between the FS topology and the coherence of low-energy quasiparticles. Indeed, for a holelike FS, coherence is suppressed at the 'hot spots' where the FS intersects the antiferromagnetic zone boundary.
When the FS turns electron-like, increased quasiparticle coherence is restored all along the FS (see Appendix for a more detailed analysis) [6][7][8][9][10][11] . At stronger coupling, several methods 9,12-28 have established that the Hubbard model displays a pseudogap which originates from antiferromagnetic correlations. These correlations become short-range as the coupling strength or doping level are increased, as found in experiments 29 . The FS topology, on the other hand, is an issue which has to do with lowenergy, long-distance physics. Hence, it is an intriguing and fundamentally important question to understand how the short-range correlations responsible for the pseudogap can be sensitive to FS changes.
Here, we study the Hubbard model for a broad range of parameters, and analyze the pseudogap and Fermi surface topology, and their interplay. We show that, at strong coupling, interactions can strongly modify the Fermi surface, making it more hole-like as compared to its non-interacting shape 16,[30][31][32][33] . We find that a pseudogap only exists when the FS is hole-like, so that p * ≤ p FS . We identify an extended parameter regime in which these two critical doping levels are very close to one another: p * p FS , so that the FS turns electron-like only when the pseudogap collapses. Moreover we show that, when considering the relation between the pseudogap and FS topology, hole-doped cuprates can be separated into two families: materials for which p * p F S and materials which have p * < p F S . These two families differ mostly by the relative magnitude of the next nearest-neighbor hopping. These findings are shown to be consistent with a large body of experiments on cuprates.
We reveal that a common link between these observa-arXiv:1707.06602v3 [cond-mat.str-el] 2 Apr 2018 tions is the pole-like feature 22,23,30,32,[34][35][36] displayed by the electronic self-energy at the antinodal point, k = (π, 0). The large imaginary part of the antinodal selfenergy associated with this pole is responsible for the pseudogap, while the large particle-hole asymmetry associated with its real part controls the interaction-induced deformation of the Fermi surface and the location of the Fermi surface topology transition. We investigate the evolution of this particle-hole asymmetry as a function of doping and nearest-neighbor hopping t , and show that the line in (p, t ) space where particle-hole symmetry is approximately obeyed at low energy is pushed, at strong coupling, to very low values of p and very negative values of t . This is in stark contrast to the results of weak-coupling theories where this line is close to the Lifshitz transition of the non-interacting system. This also explains why interactions drive the Fermi surface more hole-like for hole-doping.
In order to understand these results from a more analytic standpoint, we consider a recently developed SU (2) gauge theory of fluctuating antiferromagnetic order 37,38 ; additional results on the SU(2) gauge theory appear in a companion paper, Ref. 39. We focus on a metallic phase of 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 theory 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 provides an explanation for the quasi-pole of the physical electron self-energy. The latter is calculated and compares well to our numerical results, as do the trends in the evolution of the pseudogap and particle-hole asymmetry as a function of p and t . This paper is organized as follows. In Sec. II, we briefly introduce the model and the numerical methods used in this article. In Sec. III, we study the interplay between the pseudogap and FS topology and analyze the mechanisms controlling this interplay. The comparison and interpretation of our results in terms of the SU(2) gauge theory is presented at the end of this section. In Sec. IV we discuss the relevance of our results to experiments on hole-doped cuprates. Sec.V provides a conclusion and outlook. Finally, details about the employed methods and various supporting materials can be found in the Appendices.

II. MODEL AND METHOD
We consider the Hubbard model defined by the Hamiltonian: where U is the onsite Coulomb repulsion and µ the chemical potential. The hopping amplitudes t ij 's are chosen to be non-zero between nearest-neighbor sites (t ij = t) and next-nearest-neighbor ones (t ij = t ). These hopping amplitudes define a non-interacting dispersion relation k = −2t(cos k x + cos k y ) − 4t cos k x cos k y . In the following, t = 1 will be our unit of energy. We solve this model using two distinct methods: the dynamical cluster approximation (DCA 15 ) and determinant quantum Monte Carlo (DQMC 40 ), see the Appendix for details. Cluster extensions of dynamical mean-field theory (DMFT) have shown that the Hubbard model is able to capture many features of cuprate superconductors, such as the superconducting dome and the pseudogap 9, [14][15][16][17][18][20][21][22][23]26,32,[41][42][43] . They have also established that the pseudogap originates from antiferromagnetic correlations, which become short-range as the coupling strength or doping level are increased. This was also recently corroborated by exact diagrammatic Monte Carlo simulations 28 . While cluster extensions of DMFT have shown that hole doping can drive a Lifshitz transition [30][31][32] no general relationship between the pseudogap and FS topology has been established. We therefore carry out a systematic study for a broad range of parameters in order to investigate this issue.

A. Pseudogap and Fermi surface topology
In Fig. 1, we display the pseudogap onset temperature T * (p), and the temperature T FS (p) at which the Fermi surface changes its topology, as a function of doping level p, for several values of the next-nearest-neighbor hopping t . T * is identified as follows: we calculate the zero-frequency extrapolated value of the spectral function at the antinodal point (π, 0); we find that its temperature dependence displays a maximum which we identify as T * . Below this scale, the antinodal spectral intensity decreases, signaling the opening of a pseudogap. T FS is identified as the temperature where the Fermi surface crosses the (π, 0) point, and turns from hole-like to electron-like as temperature decreases (see below). Note that our definition of a Fermi surface is a pragmatic one: strictly speaking a Fermi surface only exists at zero temperature. At finite temperatures, we define the Fermi surface as the surface in momentum-space corresponding to the maximum of the spectral intensity as it would be observed, e.g. in an angle-resolved photoemission (ARPES) experiment 44 .
When extrapolated to zero temperature, these data define two critical doping levels: p * such that the pseudogap disappears for p > p * , and p FS that marks the transition from a hole-like FS (p < p FS ) to an electronlike FS (p > p FS ). Strikingly, the two curves in Fig. 1 suggest that the pseudogap can only exist when the Fermi surface is hole-like, i.e. that p * ≤ p FS . It appears that for values of t ≥ −0.1 both transitions happen at the same doping p * = p FS within our error bars. For more negative values of t the Fermi surface first becomes hole-like as p The finite temperature data points are extrapolated to zero temperature and yield two critical dopings p * and pFS. It is apparent that 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 estimate all uncertainties in finding T * and TFS with DCA (see also Appendix). Note that the change of topology of the Fermi surface for 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 * < p FS . We never observe a pseudogap with an electron-like Fermi surface, which would correspond to p * > p FS .
This can be documented further by repeating this analysis for several doping levels p and t values. The resulting map in the (p, t ) parameter space is displayed in Fig. 2. A first observation is that the topological transition of the FS (blue line) that separates the regions with hole-like and electron-like Fermi surfaces is strongly renormalized with respect to its non-interacting (U = 0) location (dashed line in Fig. 2 and arrows in Fig. 1). The black line defines the onset of the pseudogap. These lines define three regions: at large doping above the blue line, the FS is electron-like and no pseudogap is present. In the intermediate region between the two lines, the FS is hole-like but without a pseudogap. The topological transition and pseudogap opening coincide for a range of t , while for more negative t the two lines split apart and, as doping is reduced, the pseudogap only opens after the FS has already turned hole-like at higher doping level (p * < p FS ). The pseudogap and FS topology transition lines are dependent on the value of U . As detailed in the Appendix, a larger value of U yields a more extended regime of parameters for which p * p FS , with the 'branching point' where the two lines merge moving towards more negative values of t and larger doping level. This observation is important when comparing to experimental observations (see below).  Figure 2. Zero-temperature Fermi surface topology and pseudogap in the p−t plane. The black line separates a region with no pseudogap (no PG) from a region where a pseudogap exists (PG). The blue line indicates where the interacting Fermi surface changes its topology from electronlike (e-FS) to hole-like (h-FS). The dashed blue line signals the same transition in the non-interacting case. The red curve locates the change in particle-hole asymmetry at the antinode: above the red-line the real-part of the self-energy modifies the FS towards a more hole-like shape. On the red line, the selfenergy pole crosses zero-energy and approximate particle-hole symmetry is restored, corresponding also to a maximum of the low-energy scattering rate as t is varied for fixed p. Points A-D label a set of parameters which are discussed further in   Antinodal quasiparticle dispersion and Fermi surface topology (a) Antinodal quasiparticle energỹ (π,0) for different doping levels, as a function of temperature. The pseudogap onset temperature T * and the Lifshitz transition temperature TFS are indicated by arrows. Below the pseudogap temperature,˜ (π,0) rapidly becomes very negative, driving the FS hole-like. Only when no pseudogap is present (here for p > 0.12) does˜ (π,0) increase at low temperature and eventually becomes positive to yield an electron-like Fermi surface. (b) Illustration of the relation between the sign of (π,0) and the Fermi surface topology.

B. Change of Fermi surface topology due to correlation effects
The Fermi surface topology at the antinode is controlled by the renormalized quasiparticle energỹ For negative values of˜ (π,0) the Fermi surface is holelike, while it is electron-like for˜ (π,0) > 0. In order to gain insight in the mechanisms driving the Lifshitz transition, Fig. 3 displays˜ (π,0) as a function of temperature for various doping levels, with arrows indicating T * and T FS . Interestingly, even at the highest temperature T = 0.2 displayed there,˜ (π,0) is negative for all doping levels, yielding a hole-like Fermi surface while the non-interacting Fermi surface would be electron-like for p 9%. This temperature is above the pseudogap temperature T * , and hence the renormalization of the FS would be visible on a full Fermi surface in an ARPES experiment. In this high-temperature range, only local correlations are responsible for this effect, as already captured in a single-site DMFT calculation (see Fig. 12 in the Appendix). As temperature is decreased,˜ (π,0) first increases slightly but then suddenly drops to very negative values, pushing the Fermi surface to be very hole-like at low temperatures. This starts happening just above the pseudogap temperature and both effects can be traced back to non-local electronic correlations. For this value of t = −0.1 the connection between the disappearance of the pseudogap and the recovery of an electron-like surface is clear. Indeed, when no pseudogap is present as e.g. for p = 0.15,˜ (π,0) keeps on increasing and crosses zero, and an electron-like FS is recovered at low-T .
C. Particle-hole asymmetry and pole-like structure in the self-energy From the definition of˜ (π,0) it is clear that it is the real part of the self-energy at the antinode that drives the renormalization 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−U p/2 is the self-energy from 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 and fairly large for larger values of t . This pushes the Fermi surface topology transition to higher values of t : for 5% doping it remains hole-like up to t +0.2 whereas the Lifshitz transition of the non-interacting system occurs at t −0.05 (see also Fig. 2). The real part of the self-energy is related to its imaginary part through the Kramers-Kronig relation (3) It is therefore instructive to analyze the behavior of ImΣ (π,0) (ω) (Fig. 4b) for several values of t (as indicated by the points A, B, C and D on Fig. 2) corresponding 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, corresponding to a pole-like feature of the self-energy. For t = −0.2 (point B), this peak is centered at ω = 0. Because it is particle-hole symmetric, it leads to a vanishing real part of the self-energy (see Fig. 4a). For values of t just below and above -0.2 (points A and C), the peak in ImΣ (π,0) (ω) shifts to negative (resp. positive) values of ω. It has become particle-hole asymmetric and induces a positive (resp. negative) real part of the self-energy. There is therefore a direct connection between the existence of a large particle-hole asymmetric peak in the imaginary part of the self-energy and the renormalization of the Fermi surface to a more hole-like topology. Note that the largest value of the low-frequency scattering rate as t is varied is found when ImΣ (π,0) (ω) is particle-hole symmetric (e.g. point B in Fig. 4): this defines the location of the red line in Fig. 2 (see also the Appendix). Anywhere above this line, the self-energy is particle-hole asymmetric and drives the Fermi surface topology transition to larger doping p as compared to the non-interacting case. Note that the system becomes very incoherent below the red line, at more negative values of t and small doping. The precise nature of the Fermi surface in this region, and its possible reconstruction, is difficult to assess with the methods employed here.
This pole-like feature in the self-energy is also responsible for opening the pseudogap, as clearly seen from the inset of Fig. 4b which displays the antinodal spectral function: the minimum of the spectral intensity is found to coincide with the frequency of the quasi-pole, where ImΣ (π,0) (ω) is largest.

D. Fermi surface topology: numerically exact DQMC results
These results have been obtained using the DCA approximation with an 8-site cluster (see Appendix A). We also cross-checked these results with a different and independent method: numerically exact determinant quantum Monte Carlo (DQMC) 40 at T = 1/3. The result is displayed in Fig. 5 (left panel) and clearly shows that the antinodal self-energy drives the Fermi surface holelike over a broad region of the (p, t ) plane, in agreement with 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 with the red line of Fig. 2, and moves closer to half-filling as U is increased (see also Fig. 11) and towards the noninteracting Lifshitz transition line as U is reduced.

E. SU(2) gauge theory
Recent numerical work, using a 'fluctuation diagnostics' analysis of the contributions to the electronic self-energy in both the DCA 26 and lattice diagrammatic Monte-Carlo 28 approaches have established that the pseudogap is associated with the onset of shortrange antiferromagnetic (AF) correlations. On the analytical side, an SU(2) gauge theory approach has been introduced 37-39 to deal with states in which AF longrange order is destroyed by orientational fluctuations of the order parameter. It is thus very natural to attempt to interpret our numerical results in this framework and compare 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: In this expression, ψ ± are 'chargons' -fermions which carry charge but no spin quantum numbers and R i 's are 2 × 2 unitary matrix fields, the bosonic spinons . The R i matrix can be thought of as defining the local reference frame associated with the local AF order (for early work promoting the local reference frame to a dynamical variable, see Refs. [45][46][47]. This representation has a local gauge invariance corre- The Hubbard interaction can be decoupled using a vector field Φ i conjugate to the local spin-density c † iα σ αβ c iβ /2, and a vector 'Higgs field' is introduced such that: This identifies the Higgs field, H i , as the local antiferromagnetic moment in the rotated reference frame. Note that H i , which transforms under the adjoint of the gauge SU(2), does not carry any spin since it is invariant under a global spin rotation.
We can now consider Higgs phases in which H i = 0 but R i = 0. Because of the latter, such phases do not display long-range AF order, which has been destroyed by orientational fluctuations. However, H i = 0 signals that the local order has a non-zero amplitude. A non-zero H i also implies that such a phase has topological order, corresponding to different possible residual gauge groups once the SU(2) gauge symmetry has been spontaneously broken by the Higgs condensate [48][49][50][51] . There are different possible mean-field solutions for the Higgs condensate, U = 5 (2) = 0 (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 the region where the FS is driven hole-like becomes larger.
corresponding to different topological orders and different broken discrete symmetries 38 . Here we shall focus on the simplest one with U(1) topological order which preserves all space group, time-reversal, and spin rotations symmetries; this corresponds to the following configuration of the Higgs field (which resembles AF order): in which H 0 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 is easily calculated. Because the chargon field 'sees' an antiferromagnetic environment, it is identical to the expression obtained for an antiferromagnetic spin-density wave 39 . It thus has a matrix form which involves both components which are diagonal in momentum and offdiagonal components coupling k to k + Q: Its momentum diagonal component reads In this expression, ξ ψ k = −2Z t t(cos k x + cos k y ) − 4Z t t cos k x cos k y − µ is the renormalized dispersion of the chargons. A quantitative calculation of the renormalization factors Z t and Z t requires a full solution of the mean-field equations. We found typical values Z t ∼ 0.3 and Z t ∼ 0.2, weakly dependent on the doping level p since the chemical potential mainly affects the chargon dispersion but not the spinon dispersion. Importantly, the self-energy (8) of the chargons has a pole at ω k = ξ ψ k+Q . Hence the mean-field chargon Green's function has zeros: these zeros are located at zero energy on the Brillouin zone contour defined by ξ ψ k+Q = 0, corresponding to a chargon 'Luttinger surface'. There are two bands of chargon excitations, corresponding to the solutions of (ω − ξ ψ k )(ω − ξ ψ k+Q ) − H 2 0 = 0. To summarize, a crucial aspect of this SU(2) gauge theory description is to have chargons whose dispersions are identical (at the mean-field level) to the excitations of a spin-density wave states, despite the theory having no long-range order or broken symmetries (i.e. the symmetry is restored by the fluctuations of the spinon fields). At the mean-field level, in the phase associated with the configuration of the Higgs field considered here, the spinon excitations are gapped. In order to obtain the physical electron Green's function, a convolution of the chargon and spinon Green's function over frequency and momentum must be performed: G c = G R G ψ and the physical electron self-energy can then be obtained from Σ = ω +µ− k −G −1 c (with k the bare dispersion defined above). For the purpose of the present paper, a detailed discussion of the spinon dispersion and Green's function is 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 structure of G ψ but the location in momentum and frequency of the most singular structures of the physical self-energy are still those encoded in the chargon self-energy given by (8). (ii) The convolution does bring an important effect however: in contrast to the imaginary part of the chargon self-energy, which is constant all along the Luttinger surface ξ ψ k+Q = 0, the imaginary part of the physical electron self-energy obtained from the convolution of Green's functions has an imaginary part which is larger close to the antinodes than close to the nodes, see Fig. 15 in Appendix F. Hence, the gauge theory manages to capture qualitative aspects of the nodal-antinodal dichotomy found in our DCA calculations.
The figure also shows that the peak frequency ω p shifts from negative to positive frequency as t is increased. The inset of this figure displays the corresponding spectral function at the antinode, which has a pseudogap caused by the quasi-pole at ω p . Note that the pseudogap is particle-hole asymmetric, as expected from the fact that it does not originate from the particle-particle channel. These results are in excellent qualitative agreement with the DCA calculations above (Fig. 4). Note that, for the sake of comparison to the finite-temperature DCA results, the gauge theory calculations presented here are performed at a finite temperature larger than the spinon gap. How do gapless nodal excitations survive in the gauge theory description as temperature is lowered below this gap (e.g. by having bound-states of the chargons and spinon as in an FL* state 52 ) is an important question which is however beyond the scope of the present paper.
In Fig. 6(a), we summarize important aspects of the mean-field analysis of the gauge theory 39 as a function of doping level p and t . As in Fig. 2, the blue line in this figure is the location of the Lifshitz transition of the physical electron FS from hole to electron-like (as defined by the change of sign of the renormalized antinodal dispersion, Eq. 2) and the red line indicates where ω p = 0 (i.e. where particle-hole symmetry is approximately restored at low energy). In good qualitative agreement with the DCA results displayed in Fig. 2, one sees that the Lifshitz transition of the physical FS is pushed to much larger doping in comparison to that of the non-interacting system (dashed line), and that the location of the red line where the pole is close to zero energy is pushed to much smaller doping. The latter approximately coincides with the Lifshitz transition of the chargon Luttinger surface, given by 4Z t t = µ. Because the chemical potential µ of the interacting system takes more negative values than the non-interacting one and also because Z t < 1, the red line is shifted to lower doping as U increases, in agreement with the result of Fig. 5. This clarifies why the pole is found at positive energies for most values of (p, t ) and why the FS id driven hole-like in a wide region of the (p, t ) plane. A striking consequence of the presence of the pole is illustrated around the t = 0, p = 0 point, corresponding to the half-filled Hubbard model with only nearest-neighbor hopping, in which the antinodal scattering must be particle-hole symmetric by symmetry. When the system is very slightly hole-doped away from p = 0, both DCA and the mean-field gauge theory suggest that the particle-hole symmetric point rapidly shifts to very negative t . This is in striking contrast to weak-coupling theories in which approximate particle-hole symmetry at the antinode would be restored at the non-interacting Lifshitz transition (dashed line). We note that there are quantitative discrepancies in the location of these two lines between the numerical DCA results and the meanfield gauge theory results, which are predominantly due to the assumptions made on the renormalization parameters Z t and Z t entering the chargon dispersion and on the Higgs field amplitude H 0 .
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 the self-energy as being due to short-range antiferromagnetic correlations, long-range order being destroyed by orientational fluctuations. The quasi-pole is responsible for the pseudogap in the physical electron Green's function, while the spinon (R) spectrum displays a gap. The chargons have a spectrum characteristic of an AF spin-density wave despite the absence of AF long-range order, and their self-energy has a sharp pole at mean-field level. The (red) line where the pole crosses zero energy, corresponding to an approximate restoration of particle-hole symmetry at low-energy, can be interpreted 39 as the Lifshitz transition of the chargon Luttinger surface.

IV. DISCUSSION AND EXPERIMENTAL RELEVANCE
Our results establish that an asymmetric pole-like feature in the antinodal self-energy is responsible for both the pseudogap and for the renormalization and topological transition of the FS. We note that, in weak-coupling approaches such as spin-fluctuation theories (see Appendix E for a detailed discussion) the self-energy becomes very large for ω = k+(π,π) − µ, provided that the antiferromagnetic correlation length ξ is large enough and that v F /ξ < T . As a result, hot spots form on the Fermi surface, at specific k-vectors defined by k = k+(π,π) = µ, corresponding to the intersection of the antiferromagnetic Brillouin zone with the Fermi surface. Hence, in a weak coupling approach, the change of sign of the bare dispersion (π,0) − µ = 0 controls both the doping at which the hot spots reach the antinode and that where the Lifshitz transition occurs. As a result, the non-interacting FS transition line (blue dashed line in Fig. 2) controls at the same time the location of the Lifshitz transition, the symmetry of the self-energy and the suppression of spectral weight along the Fermi surface. This is in stark contrast to our strong-coupling results where these phenomena appear at distinct locations. In particular, we have demonstrated that the line in (p, t ) parameter space where particle-hole symmetry is approximately obeyed at low energy is pushed, at strong coupling, to very low values of p and very negative values of t , see Fig. 2 and Fig. 5 where this line is displayed in red. This is crucial in explaining why interactions drive the Fermi surface more hole-like for a wide range of (p, t ) where the non-interacting (or weak coupling) FS would actually be electron-like, and why the Lifshitz transition is pushed to larger values of p in comparison to the noninteracting system.
In order to put our results in perspective, we note that the relation between a pole-like feature in the self-energy and the pseudogap, as well as the implications of the corresponding zeros of the Green's function for the reconstruction of the Fermi surface have been previously discussed in cluster extensions of dynamical mean-field theory 22,23,30,32,[34][35][36] and in phenomenological theories The existence of a Lifshitz transition as the hole doping is increased was also discussed in some previous cluster DMFT or DCA studies [30][31][32] . However, the role played by the particle-hole asymmetry associated with the selfenergy pole in determining the FS topology, and the systematic dependence of this asymmetry on (p, t ) were not unraveled and studied, and hence the key interplay between FS topology and the pseudogap was not previously revealed.
We now discuss the relevance of our results to experiments on hole-doped cuprates. We first note that, indeed, a pseudogap is not found when the FS is electronlike and hence that the relation p * ≤ p FS is apparently obeyed in all compounds. In the single-layer compound La 2−x Sr x CuO 4 (LSCO), with a small value 57 of |t /t|, the in-plane resistivity in high magnetic fields 58 suggests that p * 0.18. Currently available ARPES experiments [59][60][61] allow to ascertain that 0.17 < p FS 0.20. In the Nd-LSCO compound, high-field transport 62 finds p * 0.23, while ARPES 63 has 0.20 < p FS < 0.24.
In another single-layer compound (Bi, Pb) 2 (Sr, La) 2 CuO 6+δ (Bi2201) [64][65][66][67] , it is found that p * p FS . An ARPES experiment on the bilayer Bi2212 material 68 has shown that the antibonding FS crosses the antinode at p FS 0.22 and suggested that it may be connected to the onset of the pseudogap. This was further confirmed in a recent electronic Raman experiment 3,4 that found the pseudogap end-point at p * 0.22. Note that the Raman response is believed to be predominantly sensitive to the antibonding band since it is close to a density of states singularity 3  , it is generally believed that p FS and p * are distinct with p * < p FS . This is in qualitative agreement with our finding that the FS and pseudogap critical doping coincide for smaller values of |t /t|, and are distinct for larger ones. Hence, we conclude on the basis of our results and experimental observations that there are two families of hole-doped cuprates: materials with smaller values of |t /t| for which the collapse of the pseudogap and change of FS topology coincide (p * p FS ), and materials with larger values of |t /t| for which these are distinct phenomena (p * < p FS ).
Finally, a very recent study on Nd-LSCO using hydrostatic pressure to tune the band structure finds that both p FS and p * decrease by the same amount. 5 This provides a compelling experimental demonstration that p * cannot exceed p FS .
We finally comment on the predicted renormalization of the FS by strong correlations. In view of Fig. 2, the materials for which this effect is expected to be strongest are the ones with smaller values of |t /t|, hence we turn to LSCO. We note that, in order to fit the ARPES FS using a single-band tight binding model, the effective parameter t has to be tuned systematically more negative (corresponding to a more negative˜ (π,0) ) as doping is reduced, i.e., from t /t = −0.12 for p = 0.3 to t /t = −0.2 for p = 0.03 60 . Moreover, electronic structure calculations based on DFT-LDA yield p FS 0.15 while ARPES finds 0.17 < p FS 0.20, as mentioned above. These two observations suggest that correlation effects indeed generally drive the FS more hole-like.

V. CONCLUSION AND OUTLOOK
To conclude, we have investigated the interplay between the pseudogap and the Fermi surface topology in the two-dimensional Hubbard model. In the weak-coupling regime these issues are directly connected: hotspots can only form when the FS is hole-like and intersects the antiferromagnetic zone boundary. At stronger coupling, the antiferromagnetic correlations responsible for the pseudogap become short-ranged, and it becomes a fundamental puzzle to understand whether there is any connection to FS topology, which is in essence longdistance physics. We provide an answer to this puzzle by showing that a common pole-like feature in the electronic self-energy controls both issues. This pole induces a large low-energy scattering rate responsible for the onset of the pseudogap, and its asymmetry leads to significant modifications of the Fermi surface with respect to its noninteracting shape and controls the location of the Lifshitz transition. As a consequence, we find that the pseudogap only appears on hole-like Fermi surfaces, i.e. p * ≤ p FS and that p * p FS for an extended range of doping levels and values of t . These findings are in good agreement with available experimental data. We have also shown that our results can be interpreted in the framework of an SU(2) gauge theory of fluctuating antiferromagnetism with topological order. This provides an explanation for the origin of the pole in the self-energy and establishes the connection between the pseudogap and the Fermi surface topology through the chargon Luttinger surface. This effort to bridge the gap between numerical results obtained within cluster extensions of DMFT and low-energy effective field theories is pursued and detailed in a companion publication 39 .
Let us emphasize that in most of the parameter range relevant to hole-doped cuprates, the self-energy pole is found at a positive energy. Hence, the strongest suppression of the antinodal spectral weight is predicted to occur at energies above the Fermi level, which is not directly accessible to ARPES experiments. While a strong particle-hole asymmetry is indeed observed by STM 74,75 , this emphasizes again 76 the importance of developing momentum-resolved spectroscopies able to probe the 'dark side' of the FS. Finally, an outstanding question is to explore whether the topological order, associated with the pseudogap regime in the gauge theory description, can be revealed more directly in numerical studies of Hubbard-like models.

ACKNOWLEDGMENTS
We are grateful to L. Taillefer and N. Doiron-Leyraud for sharing and discussing their experimental data before publication. We also acknowledge discussions with M. Civelli Our results for the two-dimensional Hubbard model are obtained using two methods: unbiased determinant quantum Monte Carlo (DQMC 40 ) and the dynamical cluster approximation (DCA 15,77 ), a cluster extension of dynamical mean-field theory (DMFT 78 ) that captures the physics of short-range spatial correlations. We perform DQMC on a 16 × 16 lattice with periodic boundary conditions at a temperature T = 1/3. Since the inverse temperature β = T −1 = 3 is significantly smaller than the linear size of the lattice L = 16, the finite size effects are negligible in the DQMC calculation. The imaginary time step was set to ∆τ = 3/64 which is small enough to avoid artifacts due to the discretization errors. We use 5.12 × 10 5 Monte Carlo sweeps to collect the data after 1000 warmup sweeps.
The DCA calculation is performed with an eight-site cluster. In the DCA approach, the lattice self-energy is approximated by a patchwise-constant self-energy Σ K in the Brillouin zone. We solved the DCA equations with an eight-site auxiliary quantum impurity cluster. In the geometry we used, the Brillouin zone is divided in eight sectors where the self-energy is constant, as shown in Fig. 7. Note that there are clearly distinct patches for the antinodal and the nodal region of the Brillouin zone.
We use both the the Hirsch-Fye 77 and the continuoustime quantum Monte Carlo 79 method to solve the auxiliary cluster impurity problem. A comparison of both methods shows that the imaginary-time step ∆τ = 1/21 (0,0) (π,0) (π,π) (0,π)  p F S f or v al u es of t gr e at er t h a n − 0 .1. F or m or e n e g ati v e v al u es of t t h e p * a n d p F S li n es s plit a p art. W h e n U is l ar g er, t his br a n c hi n g p oi nt g o es t o l o w er v al u es of t . T his is s 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 or b ot h U = 7 a n d U = 7 .5. It is cl e ar fr o m t h e fi g ur e, t h at f or U = 7 .5 p * a n d p F S ar e m u c h cl os er t h a n f or U = 7. T his c a n b e u n d ( π, 0 ) d e cr e as es si g ni fi c a ntl y a n d b e c o m es n e g ati v e. T his yi el ds 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 mp er at ur e t h at br e a ks L utti n g er's t h e or e m 8 3, 8 4 wit h a v olu 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 as e.
T h e r e d a n d bl u e li n es s h o w ˜( π, 0 ) a n d ˜( π The physical electron self-energy has clear nodal/antinodal differentiation with a stronger scattering at the antinode than at the node. Right panel: The chargon self-energy is given by Eq. 8 and has no momentum differentiation. The broadening in the chargon self-energy is η = 0.04. Appendix F: Comparison of the chargon and electron self-energy in the SU(2) gauge theory Here we illustrate in more details the role of the convolution that allows to recover the electronic Green's function in the SU(2) theory. As we have discussed above, see e.g. Fig. 6, the location in momentum and frequency of the most singular structures of the physical self-energy are not affected by the convolution and they are already encoded in the chargon self-energy given by Eq. 8. The convolution mainly smears G ψ and the electron self-energy is a broadened counterpart of the chargon self-energy. A more detailed inspection shows that the convolution also redistributes spectral weight over the Brillouin zone. As a result, the physical electron selfenergy displays nodal/antinodal differentiation, which is absent in the chargon self-energy. This is illustrated in Fig. 15 where it is clearly seen that the imaginary part of the electronic self-energy is larger close to the antinode than at the node. This differentiation is not present in the chargon self-energy.