Abstract
The pseudogap refers to an enigmatic state of matter with unusual physical properties found below a characteristic temperature T* in holedoped hightemperature superconductors. Determining T* is critical for understanding this state. Here we study the simplest model of correlated electron systems, the Hubbard model, with cluster dynamical meanfield theory to find out whether the pseudogap can occur solely because of strong coupling physics and short nonlocal correlations. We find that the pseudogap characteristic temperature T* is a sharp crossover between different dynamical regimes along a line of thermodynamic anomalies that appears above a firstorder phase transition, the Widom line. The Widom line emanating from the critical endpoint of a firstorder transition is thus the organizing principle for the pseudogap phase diagram of the cuprates. No additional broken symmetry is necessary to explain the phenomenon. Broken symmetry states appear in the pseudogap and not the other way around.
Introduction
The phase diagram of holedoped hightemperature superconductors remains puzzling. A state of matter with unusual physical properties, dubbed “the pseudogap”, is found below a characteristic temperature T* in a doped Mott insulator. Since the superconducting state is born out of the pseudogap over much of the phase diagram, the nature of the pseudogap is a fundamental issue in the field and it is under intense theoretical^{1} and experimental^{2,3,4} scrutiny.
A pseudogap can occur because of disorderbroadened longrange ordered phases of Ising type^{5} or because of fluctuating precursors to a longrange ordered phase that would appear only at T = 0 because of the MerminWagner theorem^{6}. The phenomenology of the pseudogap in the less stronglycoupled electrondoped cuprates^{7,8} differs from that of holedoped cuprates and is consistent with fluctuating precursors^{9}. The above two generic mechanisms have in common that they can occur even when interactions between electrons are not strong enough to lead to a Mott insulator at halffilling. Various broken symmetry states, such as spincharge density wave, have been linked to the pseudogap in some cuprate families^{2,3,4,9} but not in all of them. Yet the pseudogap is a generic feature of all holedoped cuprates. It is also possible that broken symmetries are different phenomena and in particular only a consequence and not the origin, of the pseudogap, as suggested recently^{10}.
On the other hand, the Mott phenomenon, a blocking of charge transport because of strong electronic repulsion, is ubiquitous in holedoped cuprates^{11} and it is appropriate to ask whether there is a third generic mechanism for the pseudogap that is associated purely with Mott physics in two dimensions^{7}. This mechanism appears only if the Coulomb repulsion is strong enough^{7} to turn the system into a Mott insulator at zero doping, as observed in holedoped cuprates. No broken translational or rotational symmetry is needed, although broken symmetry may also occur in certain cases^{5}. In previous work^{12,13}, we identified a firstorder transition at finite doping between two different metals. Here we show that the characteristic temperature T* is an unexpected example of a phenomenon observed in fluids^{14}, namely a sharp crossover between different dynamical regimes along a line of thermodynamic anomalies that appears above that firstorder phase transition, the Widom line^{14}.
Results
Widom line in doped Mott insulator
To investigate the formation of the pseudogap upon doping a Mott insulator, we study the competition between nearest neighbors hopping t and screened Coulomb repulsion U embodied in the twodimensional Hubbard model. For the cuprates, the clear momentum dependence of the selfenergy observed in photoemission (ARPES) forces one to use cluster extensions^{15,16} of dynamical meanfield theory^{17}. It is known that four sites^{18} suffices to reproduce qualitatively the ARPES spectrum. Larger cluster sizes^{19} will improve momentum resolutions and change quantitative details but should not remove firstorder transitions, as has been verified at halffilling^{20}. As in any cluster meanfield theory (dynamical or not), we can study the normalstate phase of the model by suppressing longrange magnetic, superconducting or other types of order while retaining the shortrange correlations. Previous works based on this theoretical framework have shown that a pseudogap appears close to a Mott insulator, as a result of shortrange correlations^{7,18,21,22}. But it is only with recent theoretical and computational advances^{23} that detailed results in the low temperature region of the phase diagram can be obtained.
Recently, we used cellular dynamical meanfield theory on a 2×2 plaquette to demonstrate that a firstorder transition inhabits the finite doping part of the normalstate phase diagram of the model^{12,13}. That firstorder transition occurs along a coexistence line between two types of metals and ends at a critical point (T_{p},µ_{p}) (see T − µ phase diagram in Fig. 1a). Here we show that the pseudogap is the lowdoping phase whose characteristic temperature T* can be interpreted as the Widom line.
The Widom line is defined as the line where the maxima of different thermodynamic response functions touch each other asymptotically as one approaches the critical point^{14}. The maxima become more pronounced on approaching T_{p}, diverging at T_{p}. Fig. 1b shows the chemical potential dependence of the charge compressibility κ = 1/n^{2}(dn/dµ)_{T} for several temperatures above T_{p}. Far above T_{p}, κ develops a maximum that increases and moves to higher doping with decreasing temperatures, indicating that the charge compressibility diverges at the critical point. The loci of the charge compressibility maxima, max_{µ}κ, are shown in Fig. 1a and give an estimate of the Widom line.
Crossing of the Widom line involves drastic changes in the dynamics of the system, as indicated in previous investigations on the phase diagram of fluids^{14,24}. Similarly, here we show that the pseudogap, as seen in the singleparticle density of states and in the zero frequency spin susceptibility, arises at high temperature from crossing the Widom line that radiates out of the critical point.
Identification of T* on the basis of the local density of states
First we study the development of the pseudogap in the local density of states A(ω), which can be accessed by tunnelling or photoemission spectroscopy, along paths at constant temperature or at constant doping. Figs. 2(a,c) show the evolution of the density of states with doping at a fixed temperature above and below T_{p} respectively. In the Mott insulating state, at δ = 0, the density of states consists of lower and upper Hubbard bands separated by a correlation gap. Upon hole doping, there is a dramatic transfer of spectral weight from high to low frequency, as a consequence of strong electronic correlations. The low frequency part of A(ω) develops a pseudogap, i.e. a depression in spectral weight, between a peak just below the Fermi level and a peak above the Fermi level. The pseudogap exhibits a twopeak profile that is highly asymmetric^{22}, reflecting the large particlehole asymmetry observed experimentally^{25}. Upon increasing doping, the particlehole asymmetry decreases, the spectral weight inside the pseudogap gradually fills in and the distance between the two peaks slightly decreases. Finally, at a temperaturedependent doping, the pseudogap disappears and a rather broad peak appears in the density of states which narrows with increased doping. The change from pseudogap to correlated Fermi liquid behavior occurs either by a firstorder transition when T < T_{p}, with a discontinuous change in A(ω), or by a crossover when T > T_{p}. We obtain T*(δ) from the inflection point in A(ω = 0)(δ) at finite doping (see Figs. 2(b,d)).
Figs. 2(e,g) show the temperature evolution of the density of states at constant doping. In Fig. 2(g) above δ_{p} (below µ_{p}), there is no pseudogap, only a peak around ω = 0 that broadens with increasing temperature^{13,22}. By contrast, in Fig. 2(e) below δ_{p} (above µ_{p}), the effect of increasing the temperature is to gradually fill the pseudogap, without decreasing the peaktopeak distance. We identify the disappearance of the pseudogap in the spectrum by the inflection in A(ω = 0)(T) (see Fig. 2(f) and Supplementary Fig. S1). We will discuss later the evolution with doping of T*, but the absence of pseudogap for δ > δ_{p}, already allows us to conclude that T* appears as a bridge between the Mott insulator and the firstorder transition at finite doping.
Pseudogap versus Mott insulator
Next we discuss the relationship between the pseudogap and the Mott insulator. The pseudogap is linked to Mott physics because it opens up only above the critical U for the Mott transition, in the metal near the Mott insulator. A pseudogap can occur below this threshold due to longwavelength antiferromagnetic fluctuations, but that is different physics^{6} occurring at a different energy scale. Here, only shortrange spin correlations are involved, as observed experimentally^{26} in YBa_{2}Cu_{4}O_{8} at the pseudogap temperature. The firstorder transition at finite doping, which is the terminus of the pseudogap phase, is linked as well to Mott physics because in the (U, T, µ) phase diagram, it emerges out of the Mott endpoint at halffilling and progressively moves away from halffilling with increasing U^{12,13}.
The pseudogap inherits many features from the parent Mott insulator: in both phases the electrons are bound into shortrange singlets because of the superexchange mechanism, reminiscent of the resonating valence bond state^{27}. Fig. 3a shows the temperature evolution of the probability obtained from the largest diagonal elements of the reduced density matrix on a 2 × 2 cluster^{28}, which provide direct access to spin correlations (see also Supplementary Fig. S2). Below δ_{p}, upon decreasing temperature, the singlet is gaining weight at the expense of the triplet indicating a reduction of spin fluctuations. Because of singlet formation, the spin susceptibility χ(T) drops below a characteristic temperature, as shown in Fig. 3b and as found in experiments^{29}. The inflection point in χ(T) defines a T* that moves to lower temperatures as the doping increases and approaches T_{p} as δ → δ_{p}.
Despite the magnetic behavior similar to the Mott insulator phase, the pseudogap phase is a new state of matter. At T = 0 it appears to be separated from the Mott insulator by a secondorder transition^{13}. With decreasing temperature, neither the value of the density of states at the Fermi level (Fig. 2(f)), nor the spin susceptibility (Fig. 3b), extrapolate to zero. The T* extrapolated to δ = 0 is not related to the opening of the Mott gap (Fig. 4). Finally, the peak to peak distance for the pseudogap does not extrapolate, as δ → 0, to the Mott gap, just as is observed in experiment^{30}.
T* as the Widom line
Now we move to the relationship between the pseudogap and the Widom line emanating from the critical point at finite doping. Fig. 4 shows in the T − δ plane the doping evolution of the various T*, identified above as inflection points in A(ω = 0) along constant T or constant δ paths and as inflection points in χ(T) and Prob_{singlet}(T) at constant δ. The different T* lines for the pseudogap move closer to each other along the Widom line as we approach the critical endpoint. The interrelation between T* and Widom line is our main finding. Therefore our work shows that the dynamic crossover associated with the buildup of the pseudogap is concomitant with a crossover in the thermodynamic quantities, as observed in supercritical fluids. The organizing principle of these phenomena is the Widom line. One can thus interpret T* as the Widom line, or, equivalently, consider the Widom line as a thermodynamic signal of T*. Our results suggests that all indicators (both thermodynamic and dynamic) of the pseudogap temperature scale T* should approach each other with increasing doping, joining a critical endpoint of a firstorder transition, which thereby appears as the source of anomalous behavior. In this view, pseudogap and strongly correlated Fermi liquid are separated from each other at low temperature by a firstorder transition and are thus two distinct states of matter, just as liquid and gas are two distinct states, or phases.
Discussion
Common theories to explain the pseudogap phase include the presence of rotational and/or spatial brokensymmetry phases as an essential ingredient. By contrast, in our approach the pseudogap is a consequence of large screened Coulomb repulsion that leads to strong singlet correlations in two dimensions reminiscent of resonating valence bond physics^{27}. Competing phases are not necessary to obtain a pseudogap. The pseudogap phase can however be unstable to such phases. We therefore provide a new and generic mechanism for the pseudogap in doped Mott insulators, according to which the pseudogap state is a new state of matter, whose characteristic temperature T* corresponds to the Widom line arising above a firstorder transition.
The Mott transition is often masked by brokensymmetry states. Similarly, our finitedoping transition is masked by the superconducting phase for instance^{31}. Nevertheless the rapid crossover between pseudogap and metallic phases observed above the brokensymmetry states is accessible and can be controlled by the Widom line. Such rapid change in dynamics is a hallmark of the Widom line^{14,24} and it is consistent with the strong coupling nature and the observed phenomenology of the pseudogap in the vicinity of T* in holedoped hightemperature superconductors.
From a broader perspective, our work brings the conceptual framework of the Widom line, recently developed in the context of fluids^{14,24}, to a completely different state of matter, the electronic fluid, suggesting its unexpected generality. We recall the strong impact that resulted from bringing in the field of electronic properties of solids the well known concepts of smectic and nematic order developed earlier in the field of liquid crystals. It is tempting to argue that the same fate awaits the Widom line.
Methods
Our results are based on the cellular dynamical meanfield theory (CDMFT)^{15,16} solution of the twodimensional Hubbard model on the square lattice,
where c_{iσ} and operators destroy and create electrons on site i with spin σ and . t is the hopping amplitude between nearest neighbors, U the energy cost of double occupation at each site and µ the chemical potential. CDMFT isolates a cluster of lattice sites, here a 2×2 plaquette and replaces the missing lattice environment by a bath of noninteracting electrons which is selfconsistently determined. The cluster in a bath problem is solved by a continuoustime Quantum Monte Carlo summation of all diagrams obtained by expanding the partition function in powers of the hybridization between bath and cluster^{23,28}. The size of the plaquette is large enough to be consistent with the experimental observation^{26} that at T*, in holedoped cuprates, the antiferromagnetic correlation length is one or two lattice spacings.
The value of Coulomb interaction U = 6.2 t is larger than the critical threshold U_{MIT} ≈ 5.95 t necessary to obtain a Mott insulator at halffilling^{12,13} and is chosen such that the pseudogap critical temperature T_{p}(U) is accessible with our method. We carry out simulations at constant temperature for several values of µ and at constant doping δ = 1−n for several temperatures. Critical slowing down is a widespread and standard signal that the system is approaching a critical threshold^{32} and appears in our simulations along the Widom line close to the critical point. To obtain reliable results, we perform up to 10^{7} Monte Carlo sweeps, averaged over 64 processors and hundreds of CDMFT iterations. The typical error on n is of order 10^{−5}.
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Acknowledgements
We thank E. Kats and L. Taillefer for discussions. This work was partially supported by FQRNT, by the Tier I Canada Research Chair Program (A.M.S.T.) and by NSF DMR0746395 (K.H.). Computational resources were provided by CFI, MELS, the RQCHP, Calcul Québec and Compute Canada. A.M.S.T is grateful to the Harvard Physics Department for support and P.S. for hospitality during the writing of this work. Partial support was also provided by the MITHarvard Center for Ultracold Atoms.
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G.S. conceived the project and carried the data analysis. P.S. and K.H. wrote the main codes. G.S. and A.M.S.T. wrote the paper and all authors discussed it. A.M.S.T. supervised the entire project.
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Sordi, G., Sémon, P., Haule, K. et al. Pseudogap temperature as a Widom line in doped Mott insulators. Sci Rep 2, 547 (2012). https://doi.org/10.1038/srep00547
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