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

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

Wei Wu Centre de Physique Théorique, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France    Mathias S. Scheurer Department of Physics, Harvard University, Cambridge MA 02138, USA    Shubhayu Chatterjee Department of Physics, Harvard University, Cambridge MA 02138, USA    Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138, USA Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5 Department of Physics, Stanford University, Stanford, CA 94305    Antoine Georges Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth avenue, New York, NY 10010, USA Centre de Physique Théorique, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France DQMP, Université de Genève, 24 quai Ernest Ansermet, CH-1211 Genève, Suisse    Michel Ferrero Centre de Physique Théorique, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
Abstract

One of the distinctive features of hole-doped cuprate superconductors is the onset of a ‘pseudogap’ below a temperature . 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 high-temperature superconductors is the nature of the ’pseudogap’ Norman et al. (2005); Alloul (2014) in their phase diagram. Below a temperature which is a decreasing function of the hole-doping level , a pseudogap develops, corresponding to a suppression of low-energy excitations apparent in many experimental probes. Extrapolated to zero-temperature, defines a critical hole doping above which the pseudogap disappears as doping is increased. Another important critical value of the doping, denoted here , is that at which the Fermi surface topology changes from hole-like to electron-like, corresponding to a Lifshitz transition. Recent experiments on BiSrCaCuO  (Bi2212) have suggested that the pseudogap may be very sensitive to the Fermi surface (FS) topology and that in this compound Benhabib et al. (2015); Loret et al. (2017). In a simultaneous and independent manner from the present theoretical work, Doiron-Leyraud et al. Doiron-Leyraud et al. (2017) recently performed a systematic experimental study using hydrostatic pressure as a control parameter in the (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 weak-coupling scenarios of pseudogap physics, there is a natural connection between the FS topology and the coherence of low-energy quasiparticles. Indeed, for a hole-like 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) Honerkamp et al. (2001); Abanov et al. (2003); Löhneysen et al. (2007); Tremblay et al. (2006); Metlitski and Sachdev (2010); Montiel et al. (2017). At stronger coupling, several methods Dagotto et al. (1992); Preuss et al. (1997); Lichtenstein and Katsnelson (2000); Maier et al. (2005); Civelli et al. (2005); Macridin et al. (2006); Kyung et al. (2006); Tremblay et al. (2006); Scalapino (2007); Haule and Kotliar (2007); Ferrero et al. (2009); Gull et al. (2009, 2010); Sordi et al. (2011, 2012); Gunnarsson et al. (2015); Fratino et al. (2016); Wu et al. (2017) 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 Kastner et al. (1998). The FS topology, on the other hand, is an issue which has to do with low-energy, 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 Maier et al. (2002); Civelli et al. (2005); Chen et al. (2012); Sakai et al. (2009); Tocchio et al. (2012). We find that a pseudogap only exists when the FS is hole-like, so that . We identify an extended parameter regime in which these two critical doping levels are very close to one another: , 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 and materials which have . 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 observations is the pole-like feature Maier et al. (2002); Stanescu and Kotliar (2006); Berthod et al. (2006); Gull et al. (2009); Sakai et al. (2009); Gull et al. (2010); Lin et al. (2010) displayed by the electronic self-energy at the antinodal point, . The large imaginary part of the antinodal self-energy 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 , and show that the line in space where particle-hole symmetry is approximately obeyed at low energy is pushed, at strong coupling, to very low values of and very negative values of . 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 Sachdev et al. (2009); Chatterjee et al. (2017); additional results on the SU(2) gauge theory appear in a companion paper, Ref. Scheurer et al., 2017. 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 and .

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:

(1)

where is the onsite Coulomb repulsion and the chemical potential. The hopping amplitudes ’s are chosen to be non-zero between nearest-neighbor sites () and next-nearest-neighbor ones (). These hopping amplitudes define a non-interacting dispersion relation . In the following, will be our unit of energy. We solve this model using two distinct methods: the dynamical cluster approximation (DCA Maier et al. (2005)) and determinant quantum Monte Carlo (DQMC Blankenbecler et al. (1981)), 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 Lichtenstein and Katsnelson (2000); Jarrell et al. (2001); Maier et al. (2005); Civelli et al. (2005); Macridin et al. (2006); Kyung et al. (2006); Tremblay et al. (2006); Haule and Kotliar (2007); Ferrero et al. (2009); Werner et al. (2009); Gull et al. (2009); Sakai et al. (2009); Gull et al. (2010, 2013); Gunnarsson et al. (2015). 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 Wu et al. (2017). While cluster extensions of DMFT have shown that hole doping can drive a Lifshitz transition Maier et al. (2002); Sakai et al. (2009); Chen et al. (2012) 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.

Iii Results

iii.1 Pseudogap and Fermi surface topology

Figure 1: Pseudogap and Lifshitz transition temperatures. Evolution of the pseudogap onset temperature (black) and the Lifshitz transition temperature (blue) as a function of the hole doping , for several values of and . The finite temperature data points are extrapolated to zero temperature and yield two critical dopings and . It is apparent that for and , while for . The solid lines are linear (for ) and quadratic (for ) least squares fits to the data points, except the line of where collapses to zero close to . Error bars estimate all uncertainties in finding and 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).

In Fig. 1, we display the pseudogap onset temperature , and the temperature at which the Fermi surface changes its topology, as a function of doping level , for several values of the next-nearest-neighbor hopping . is identified as follows: we calculate the zero-frequency extrapolated value of the spectral function at the antinodal point (); we find that its temperature dependence displays a maximum which we identify as . Below this scale, the antinodal spectral intensity decreases, signaling the opening of a pseudogap. is identified as the temperature where the Fermi surface crosses the 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 Damascelli et al. (2003).

When extrapolated to zero temperature, these data define two critical doping levels: such that the pseudogap disappears for , and that marks the transition from a hole-like FS () to an electron-like 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 . It appears that for values of both transitions happen at the same doping within our error bars. For more negative values of the Fermi surface first becomes hole-like as is reduced, and the pseudogap opens at a lower doping, i.e. . We never observe a pseudogap with an electron-like Fermi surface, which would correspond to .

Figure 2: Zero-temperature Fermi surface topology and pseudogap in the 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 electron-like (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 self-energy pole crosses zero-energy and approximate particle-hole symmetry is restored, corresponding also to a maximum of the low-energy scattering rate as is varied for fixed . Points A-D label a set of parameters which are discussed further in Fig. 4.

This can be documented further by repeating this analysis for several doping levels and values. The resulting map in the 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 () 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 , while for more negative 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 (). The pseudogap and FS topology transition lines are dependent on the value of . As detailed in the Appendix, a larger value of yields a more extended regime of parameters for which , with the ‘branching point’ where the two lines merge moving towards more negative values of and larger doping level. This observation is important when comparing to experimental observations (see below).

iii.2 Change of Fermi surface topology due to correlation effects

The Fermi surface topology at the antinode is controlled by the renormalized quasiparticle energy

(2)

For negative values of the Fermi surface is hole-like, while it is electron-like for . In order to gain insight in the mechanisms driving the Lifshitz transition, Fig. 3 displays as a function of temperature for various doping levels, with arrows indicating and . Interestingly, even at the highest temperature displayed there, is negative for all doping levels, yielding a hole-like Fermi surface while the non-interacting Fermi surface would be electron-like for . This temperature is above the pseudogap temperature , 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, 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 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 , keeps on increasing and crosses zero, and an electron-like FS is recovered at low-.

Figure 3: Antinodal quasiparticle dispersion and Fermi surface topology (a) Antinodal quasiparticle energy for different doping levels, as a function of temperature. The pseudogap onset temperature and the Lifshitz transition temperature are indicated by arrows. Below the pseudogap temperature, rapidly becomes very negative, driving the FS hole-like. Only when no pseudogap is present (here for ) does increase at low temperature and eventually becomes positive to yield an electron-like Fermi surface. (b) Illustration of the relation between the sign of and the Fermi surface topology.

iii.3 Particle-hole asymmetry and pole-like structure in the self-energy

From the definition of 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 and display as a function of , in which is the self-energy from which the Hartree (infinite frequency) contribution has been subtracted out. It is seen that changes sign around and becomes negative and fairly large for larger values of . This pushes the Fermi surface topology transition to higher values of : for 5% doping it remains hole-like up to whereas the Lifshitz transition of the non-interacting system occurs at (see also Fig. 2).

Figure 4: Evolution of the antinodal self-energy at fixed doping , as a function of . (a): Real (Hartree subtracted, see text) and imaginary parts of the antinodal self-energy at . The real part vanishes where the imaginary part is maximum, corresponding to a particle-hole symmetric low-energy . (b): Real-frequency scattering rate obtained from the Maximum Entropy method for different values of . It displays a pole-like feature that crosses zero at (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 more hole-like. Inset: antinodal spectral function at point D at . See Fig. 2 for the locations of points in the plane.

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 (Fig. 4b) for several values of (as indicated by the points A, B, C and D on Fig. 2) corresponding to positive, vanishing and negative values of . 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 (point B), this peak is centered at . Because it is particle-hole symmetric, it leads to a vanishing real part of the self-energy (see Fig. 4a). For values of just below and above -0.2 (points A and C), the peak in shifts to negative (resp. positive) values of . 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 is varied is found when 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 as compared to the non-interacting case. Note that the system becomes very incoherent below the red line, at more negative values of 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 is largest.

Figure 5: Particle-hole asymmetry from determinant quantum Monte Carlo (see methods). (a): Real part of the self-energy in the plane for . For a broad region (indicated in red), is negative, hence driving the Fermi surface more hole-like, in agreement with our DCA results. (b): The line where and where the antinodal scattering rate is largest is indicated for different values of (to be compared to the red line in Fig.  2). As is increased the region where the FS is driven hole-like becomes larger.

iii.4 Fermi surface topology: numerically exact DQMC results

These results have been obtained using the DCA approximation with an -site cluster (see Appendix A). We also cross-checked these results with a different and independent method: numerically exact determinant quantum Monte Carlo (DQMC) Blankenbecler et al. (1981) at . The result is displayed in Fig. 5 (left panel) and clearly shows that the antinodal self-energy drives the Fermi surface hole-like over a broad region of the plane, in agreement with our DCA calculations. One can again observe a line where vanishes, mapped out for several values of in the right panel. This line compares with the red line of Fig. 2, and moves closer to half-filling as is increased (see also Fig. 11) and towards the non-interacting Lifshitz transition line as is reduced.

iii.5 SU(2) gauge theory

Recent numerical work, using a ‘fluctuation diagnostics’ analysis of the contributions to the electronic self-energy in both the DCA Gunnarsson et al. (2015) and lattice diagrammatic Monte-Carlo Wu et al. (2017) approaches have established that the pseudogap is associated with the onset of short-range antiferromagnetic (AF) correlations. On the analytical side, an SU(2) gauge theory approach has been introduced Sachdev et al. (2009); Chatterjee et al. (2017); Scheurer et al. (2017) to deal with states in which AF long-range 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 :

(4)

In this expression, are ‘chargons’ - fermions which carry charge but no spin quantum numbers and ’s are unitary matrix fields, the bosonic spinons (). The 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. Shraiman and Siggia, 1988; Schulz, 1990; Schrieffer, 2004). This representation has a local gauge invariance corresponding to , with an SU(2) matrix. The Hubbard interaction can be decoupled using a vector field conjugate to the local spin-density , and a vector ‘Higgs field’ is introduced such that:

(5)

This identifies the Higgs field, , as the local antiferromagnetic moment in the rotated reference frame. Note that , 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 but . Because of the latter, such phases do not display long-range AF order, which has been destroyed by orientational fluctuations. However, signals that the local order has a non-zero amplitude. A non-zero 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 Fradkin and Shenker (1979); Read and Sachdev (1991); Wen (1991); Bais et al. (1992). There are different possible mean-field solutions for the Higgs condensate, corresponding to different topological orders and different broken discrete symmetries Chatterjee et al. (2017). 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):

(6)

in which is the Higgs field amplitude and .

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 waveScheurer et al. (2017). It thus has a matrix form which involves both components which are diagonal in momentum and off-diagonal components coupling to :

(7)

Its momentum diagonal component reads

(8)

In this expression, is the renormalized dispersion of the chargons. A quantitative calculation of the renormalization factors and requires a full solution of the mean-field equations. We found typical values and , weakly dependent on the doping level 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 . Hence the mean-field chargon Green’s function has zeros: these zeros are located at zero energy on the Brillouin zone contour defined by , corresponding to a chargon ‘Luttinger surface’. There are two bands of chargon excitations, corresponding to the solutions of . 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: and the physical electron self-energy can then be obtained from (with 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. Scheurer et al., 2017 for details. It is sufficient here to emphasize the two following points. (i) The convolution mainly broadens the pole structure of 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 , 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 shifts from negative to positive frequency as 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 . 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 Senthil et al. (2003)) 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 Scheurer et al. (2017) as a function of doping level and . 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 (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 . Because the chemical potential of the interacting system takes more negative values than the non-interacting one and also because , the red line is shifted to lower doping as increases, in agreement with the result of Fig. 5. This clarifies why the pole is found at positive energies for most values of and why the FS id driven hole-like in a wide region of the () plane. A striking consequence of the presence of the pole is illustrated around the 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 , both DCA and the mean-field gauge theory suggest that the particle-hole symmetric point rapidly shifts to very negative . 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 mean-field gauge theory results, which are predominantly due to the assumptions made on the renormalization parameters and entering the chargon dispersion and on the Higgs field amplitude .

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 interpretedScheurer et al. (2017) as the Lifshitz transition of the chargon Luttinger surface.

Figure 6: Pseudogap and Fermi surface topology within the SU(2) gauge theory at mean-field level. Color coding is identical to Fig. 2 (DCA results), to which this figure should be compared. (a) Solid blue line: Lifshitz transition of the interacting Fermi surface. Along the red line, the self-energy pole is at zero energy and approximate particle-hole symmetry is restored. This also corresponds to the Lifshitz transition of the chargons. (b) Electronic self-energy at the antinode for two different values of . The quasi-pole in the self-energy moves from negative to positive frequency as is increased, see Fig. 4. The inset displays the antinodal spectral function for , emphasizing that its minimum coincides with the position of the pole. Here we assumed , a spinon gap , and (the nearest-neighbor coupling of the spin-wave fluctuations, see Ref. Scheurer et al., 2017). A broadening factor is used to obtain smooth spectral functions.

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 , provided that the antiferromagnetic correlation length is large enough and that . As a result, hot spots form on the Fermi surface, at specific -vectors defined by , 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 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 parameter space where particle-hole symmetry is approximately obeyed at low energy is pushed, at strong coupling, to very low values of and very negative values of , 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 where the non-interacting (or weak coupling) FS would actually be electron-like, and why the Lifshitz transition is pushed to larger values of in comparison to the non-interacting 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 Maier et al. (2002); Stanescu and Kotliar (2006); Berthod et al. (2006); Gull et al. (2009); Sakai et al. (2009); Gull et al. (2010); Lin et al. (2010) and in phenomenological theories such as YRZ Yang et al. (2006) or other approaches Sakai et al. (2016); Dave et al. (2013) (see Ref. Qi and Sachdev, 2010 for a gauge-theory perspective on the YRZ phenomenology). The existence of a Lifshitz transition as the hole doping is increased was also discussed in some previous cluster DMFT or DCA studies Maier et al. (2002); Sakai et al. (2009); Chen et al. (2012). However, the role played by the particle-hole asymmetry associated with the self-energy pole in determining the FS topology, and the systematic dependence of this asymmetry on 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 electron-like and hence that the relation is apparently obeyed in all compounds. In the single-layer compound (LSCO), with a small value Pavarini et al. (2001) of , the in-plane resistivity in high magnetic fields Cooper et al. (2009) suggests that . Currently available ARPES experiments Chang et al. (2008); Yoshida et al. (2006); Park et al. (2013) allow to ascertain that . In the Nd-LSCO compound, high-field transport Collignon et al. (2016) finds , while ARPES Matt et al. (2015) has .

In another single-layer compound (Bi2201) He et al. (2014); Zheng et al. (2005); Kawasaki et al. (2010); Kondo et al. (2004), it is found that . An ARPES experiment on the bilayer Bi2212 material Kaminski et al. (2006) has shown that the antibonding FS crosses the antinode at and suggested that it may be connected to the onset of the pseudogap. This was further confirmed in a recent electronic Raman experiment Benhabib et al. (2015); Loret et al. (2017) that found the pseudogap end-point at . 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 Benhabib et al. (2015) and does not give information about the possible existence of a pseudogap in the bonding band (which remains hole-like for all dopings). In compounds with larger values Pavarini et al. (2001) of , such as  Hossain et al. (2008); Badoux et al. (2016),  Platé et al. (2005); Proust et al. (2002) or  Tabis et al. (2014), it is generally believed that and are distinct with . This is in qualitative agreement with our finding that the FS and pseudogap critical doping coincide for smaller values of , 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 for which the collapse of the pseudogap and change of FS topology coincide (), and materials with larger values of for which these are distinct phenomena ().

Finally, a very recent study on Nd-LSCO using hydrostatic pressure to tune the band structure finds that both and decrease by the same amount. Doiron-Leyraud et al. (2017) This provides a compelling experimental demonstration that cannot exceed .

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 , 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 has to be tuned systematically more negative (corresponding to a more negative ) as doping is reduced, i.e., from for to for  Yoshida et al. (2006). Moreover, electronic structure calculations based on DFT-LDA yield while ARPES finds , 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: hot-spots 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 long-distance 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 non-interacting 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. and that for an extended range of doping levels and values of . 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 Scheurer et al. (2017).

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 Kohsaka et al. (2007); Fischer et al. (2007), this emphasizes again Sakai et al. (2013) 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.

Acknowledgements.
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, M. Kim, G. Kotliar, A. J. Millis, O. Parcollet, I. Paul, A. Sacuto and A.-M.S. Tremblay. This work was supported by the Simons Foundation Many-Electron Collaboration, the European Research Council (project ERC-319286-‘QMAC’), the Swiss National Supercomputing Centre (CSCS, project s575), the NSF under 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 Institute (SS) is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. MS acknowledges support from the German National Academy of Sciences Leopoldina through grant LPDS 2016-12. SS acknowledges support from Cenovus Energy at Perimeter Institute, and from the Hanna Visiting Professor program at Stanford University. AG and SS are grateful for the stimulating atmosphere of the May 2017 Jouvence workshop.

Appendix A Methods

Our results for the two-dimensional Hubbard model are obtained using two methods: unbiased determinant quantum Monte Carlo (DQMC Blankenbecler et al. (1981)) and the dynamical cluster approximation (DCA Maier et al. (2005); Hirsch and Fye (1986)), a cluster extension of dynamical mean-field theory (DMFT Georges et al. (1996)) that captures the physics of short-range spatial correlations. We perform DQMC on a lattice with periodic boundary conditions at a temperature . Since the inverse temperature is significantly smaller than the linear size of the lattice , the finite size effects are negligible in the DQMC calculation. The imaginary time step was set to which is small enough to avoid artifacts due to the discretization errors. We use Monte Carlo sweeps to collect the data after 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 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.

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

We use both the the Hirsch-Fye Hirsch and Fye (1986) and the continuous-time quantum Monte Carlo Rubtsov et al. (2005) method to solve the auxiliary cluster impurity problem. A comparison of both methods shows that the imaginary-time step used in the Hirsch-Fye solver is small enough, as shown in the Appendix B. We use 50 DMFT iterations to get converged result and use Monte Carlo sweeps at each iteration. In order to have better statistics, the results are averaged over the last few converged iterations. The typical statistical error in the real part of the self-energy and the spectral intensity at zero frequency is .

We identify the pseudogap temperature as the maximum of the temperature dependent spectral weight . It is obtained from a linear extrapolation to zero frequency of at the first two Matsubara frequencies. We find from the zero of the effective dispersion at the antinode .

Finally, the real-frequency spectral function and the self-energy are found with the maximum entropy analysis Bergeron and Tremblay (2016); Jarrell and Gubernatis (1996) on the Green’s function and the self-energy . We use two independent maximum entropy codes Bergeron and Tremblay (2016); Jarrell and Gubernatis (1996) to make sure that their results agree.

Appendix B Supporting material for the results shown in the main text

b.1 Pseudogap onset temperature and Fermi surface topology transition temperature

We identify the pseudogap onset temperature as the temperature where the spectral function at the antinodal point reaches a maximum as temperature is lowered, see Fig. 8. The zero-frequency value of the spectral function which is obtained by a linear extrapolation of the value of (the result of the numerical calculation) at its first two Matsubara frequencies. We found that different approximations of the spectral function, such as  Trivedi and Randeria (1995) yield the same values of . Also, we have used different imaginary time discretization steps in the Hirsch-Fye algorithm and observe that the results are the same for all values of . We have used throughout our work.

Figure 8: as a function of temperature . The maximum is identified as the pseudogap onset temperature . Upper panel: for and different doping levels. Lower panel: for and different discrete time steps of the Hirsch-Fye impurity solver. We can see that is already converged for .
Figure 9: The effective quasiparticle dispersion at the antinode as a function of temperature for two doping levels. At the Fermi surface topology transition temperature , we have . We show results using both a continuous-time QMC Rubtsov et al. (2005) and a Hirsch-Fye Hirsch and Fye (1986) impurity solver (the latter with a finite imaginary-time step ).
Figure 10: The real part of the antinodal self-energy as a function of temperature for different dopings and . Note that the constant Hartree shift has been removed. Left panel: For , at , is positive at low temperatures while at it is negative, suggesting is between 0 and 0.01. Central panel: For , we have . Right panel: For , we find . Therefore for fixed , increasing drives closer to the 0 (half-filling).

When the Fermi surface crosses it undergoes a Lifshitz transition and changes from hole-like to electron-like. We define the Fermi surface at finite temperature by the location of the maximum spectral intensity at zero energy (as seen in an ARPES experiment). This maximum goes through when the quasiparticle effective dispersion at the antinode as discussed in the main text (see Equation 1). We can therefore find the Fermi surface topology transition temperature by finding the zero of as a function of temperature as shown in Fig. 9. We also display the results as obtained using two different impurity solvers and show that they give identical results.

b.2 Maximum low-energy scattering line and particle-hole asymmetry

As we have shown in the main text, there is a curve in the diagram that separates a region where the pole-like feature in the imaginary part of the antinodal self-energy is on the negative energy side from a region where it is on the positive energy side. When the pole is on the positive energy side, the real part of the self-energy (with the Hartree term removed) is negative, while it is positive when the pole is on the negative side. We can therefore locate the curve by finding, at fixed , the value of the doping at which the zero-temperature extrapolation of changes sign see Fig. 10.

b.3 dependence of the connection between and

In the main text, all calculations have used . For this value, we have shown that, for values of greater than . For more negative values of the and lines split apart. When is larger, this branching point goes to lower values of . This is shown in Fig. 11 where we compute and for both and . It is clear from the figure, that for and are much closer than for . This can be understood because a larger value of extends the pseudogap region to larger dopings, while the Fermi surface topology is not influenced much by correlations before we actually have a pseudogap and hence .

Figure 11: and as a function of doping at , (dashed lines), and (solid lines). Note that here the imaginary-time discretization step is .

Appendix C Temperature evolution of and role of non-local correlations

In Fig. 12, we investigate the role of non-local correlations by comparing the results obtained by DCA, as in the main text, and single-site dynamical mean-field (DMFT) that only accounts for local correlations. The black line show as computed by DMFT for and . For these parameters, the non-interacting Fermi surface is electron-like. It is seen that at low temperature the DMFT results also predict an electron-like Fermi surface. This is not surprising as DMFT preserves the Luttinger theorem and the interacting Fermi surface is the same as the non-interacting one when . However, as temperature is increased, decreases significantly and becomes negative. This yields a hole-like interacting Fermi surface at high temperature that breaks Luttinger’s theorem Deng et al. (2013); Shastry (2011) with a volume larger than in the non-interacting case.

The red and blue lines show and respectively as obtained by DCA. has been shifted by a constant that corresponds to the energy difference of the non-interacting dispersion at and . At high temperatures all curves yield the same value, compatible with a self-energy that is essentially local. As temperature is decreased the nodal behaves like the DMFT solution indicating that the Fermi surface at the node is very close to its non-interacting shape. The DCA has a different behavior. As temperature is lowered it quickly departs from showing the onset of nodal/antinodal differentiation. At a temperature slightly above , non-local correlations become large and induce a very negative as discussed in the main text.

Figure 12: Temperature dependence of as obtained by DCA and DMFT at hole doping and .

Appendix D Pseudogap and Fermi surface topology transition as a function of

Our results for show that, for a broad range of parameters, the pseudogap disappears at the same critical doping where the Fermi surface undergoes a Lifshitz transition. In Fig. 13, we investigate how our results depend on the correlation strength . It is shown that for values of correlations have little effect on the Fermi surface topology and the low-energy scattering rate is very small. Above correlation effects set in quickly as shown by a fast increase in the value of . This induces a pseudogap at . At the same time the effective quasiparticle dispersion crosses zero and becomes very negative for larger values of . This sudden increase of the correlation effects for might explain why the pseudogap and the Fermi surface topology happen at the same time.

Figure 13: Correlation effects shown as a function of for three quantities: the quasiparticle effective dispersion , the imaginary part of the antinodal self-energy at zero energy and the difference in spectral intensity at the Fermi level for the two lowest calculated temperatures indicating whether a pseudogap has formed.

Appendix E Comparison with weak-coupling approaches

Let us investigate how our results differ from weak-coupling approaches, such as spin-fluctuation theory or the two-particle self-consistent approach (TPSC) of Vilk and Tremblay Vilk and Tremblay (1997). The latter has been shown to be quite accurate in the weak to intermediate coupling regime of the two-dimensional Hubbard model, and this appendix closely follows the analysis in Ref. Vilk and Tremblay, 1997.

In those approaches, the self-energy is obtained as

(9)

where is the non-interacting Green’s function, is the spin susceptibility, and is a coupling constant with the dimension of energy. When the magnetic correlation length is large, can be approximated by: Tremblay (2012)

(10)

with the antiferromagnetic wave-vector. The self-energy thus reads

(11)

In the regime of interest here (large enough , renormalized classical regime), the above sum is dominated by the smallest Matsubara frequency (note that ) and one obtains the imaginary part of the retarded real-frequency self-energy in the form:

(12)

The important point is that in two dimensions, this integral diverges as the correlation length becomes large, which leads to the formation of ‘hot spots’ at which a pseudogap opens. This integral can actually be performed analytically, and one finally obtains:

(13)

where ‘reg.’ denotes a non-singular contribution. The physics associated with a weak-coupling description of spin fluctuations can be entirely described on the basis of this expression  Vilk and Tremblay (1997). Let us focus first on the Fermi surface properties, corresponding to and momenta such that . As clear from (13), the self-energy is regular on the Fermi surface except at the ‘hot spots’ satisfying also , corresponding to the intersection of the Fermi surface with the antiferromagnetic Brillouin zone. At these hot-spots, the self-energy is singular: its imaginary part is of order:

(14)

This is large only when the correlation length is large: . In this regime, spectral weight is strongly depleted at the hot spots, corresponding to the weak-coupling description of the pseudogap. When the correlation length remains finite as temperature is lowered (short-range order), the hot spots and corresponding pseudogap disappear for and conventional Fermi liquid behavior is recovered at low temperature.

Let us emphasize the crucial differences that exist between the weak-coupling expression of the self-energy (13) and both the self-energy that we obtain from DCA at strong coupling, as well as the self-energy obtained from the SU(2) gauge theory. As clear from (13), the imaginary part of the weak-coupling self-energy does display a peak, but (i) the height of this peak is proportional to and thus eventually the peak and the hot spots disappear at low-T if remains finite (ii) the width of this peak is proportional to , which in the regime where the peak exists is smaller than temperature. In contrast, in the strong-coupling DCA calculations the peak is not suppressed as is reduced, and its width is larger than . Furthermore, the correlation length that we can estimate in our DCA results from the static staggered susceptibility is quite small at strong coupling: we find for example: for at . The weak-coupling expression also has a different structure than the singular delta-function form of the chargon self-energy in the SU(2) gauge theory: the latter, importantly, does not involve the correlation length (set by the spinons) and is similar to that of an SDW in the ordered phase.

Figure 14: The lines in the plane show where the low-energy imaginary part of the antinodal self-energy has its pole-like feature centered around and an essentially particle-hole symmetric low-energy spectrum. On these curves vanishes. The solid lines are obtained by DQMC for different values of , while the dashed line is the result from the SDW weak-coupling approach. The dashed line coincides with the non-interacting Lifshitz transition.

The real part of the self-energy corresponding to (13) can be obtained using Kramers-Kronig relations as:

(15)

in which we have used the short-hand notation . In the temperature regime where hot spots are present , one can distinguish two regimes of frequencies. For , is of order , while at low frequencies , the self-energy is regular . Hence, the regular part of the self-energy is regular at low-frequency even close to the hot spots when remains finite. As a result, hot spots exist at the intersection between the Fermi surface and the antiferromagnetic Brillouin zone for , but there is no reconstruction of the Fermi surface otherwise. Hence, for a doping value larger than the value corresponding to the non-interacting Lifshitz transition, there are no hot spots and the Fermi surface is weakly renormalized and electron-like. Hence, in weak-coupling the non-interacting Lifshitz transition controls both the location of the self-energy singularities and the topological transition of the Fermi surface. This is very different from our results in the strong coupling regime where these phenomena are controlled by three different lines.

By varying one can observe how the transition from weak to strong coupling happens. Fig. 14 shows the DQMC results for several values of . The lines show where the real part of the self-energy vanishes. It separates a region where the pole in the self-energy is at negative energies and one where it is on the positive side. It is seen that as becomes smaller the lines slowly approach the non-interacting Lifshitz transition, as expected in weak-coupling.

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 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 self-energy 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.

Figure 15: Scattering rate of the electrons and chargons in momentum space. The parameters used here are: . Left panel: 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 .

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