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FEATURE ARTICLE HIGH-TEMPERATURE SUPERCONDUCTIVITY FROM SHORT-RANGE REPULSION BY ANDRÉ-MARIE S. TREMBLAY S tandard solid-state physics books offer limited help to understand much of the phenomenology of high-temperature superconductors. Almost every basic assumption behind the theories in these books fails. For example, one expects the true elementary excitations of a metal to have the same quantum numbers as free electrons, in other words one expects the so-called Fermi liquid theory to apply. That is not the case for the cuprates. One instead finds that at half-filling they are insulators and that, lightly-doped away from half-filling, their metallic state at finite temperature is not a Fermi liquid. Another tenet of solid state physics is that strong Coulomb repulsion is detrimental to superconductivity. Apparently not so for the cuprates since the superconducting transition temperature is high despite the strong Coulomb repulsion necessary to explain their insulating behavior at half-filling. The challenge, like in many other frontier fields of physics, is to understand strong interactions. Much ingenuity and hard work has been invested towards this objective. The paper by Arun Paramekanti in this issue of Physics in Canada explains in simple terms how strong correlations lead to an insulator at half-filling. This phenomenon is called the Mott transition. It also describes some of the methods that have been developed to understand the whole phase diagram. Here I focus on two recently developed methods that involve numerical computations, and on what we learn from them. I outline mainly the contribution from our group in Sherbrooke, so there is definitely some bias. I begin by describing the model. SUMMARY Theoretical models of high temperature superconductors cannot be treated by perturbation theory. Here I briefly survey some of the new theoretical methods that have been developed for the Hubbard model and the insights they have provided to understand the Physics of strong interactions and high-temperature superconductors. THE MODEL, WEAK AND STRONG COUPLING Simple models in physics can have surprisingly rich sets of solutions that explain observed complex behavior with minimal assumptions. The model proposed by John Hubbard in 1963 [1] fits this category. Anderson [2] proposed it for high-temperature superconductivity right after its discovery in 1986. It contains two terms, one for kinetic energy in a single band, and the other for shortrange Coulomb repulsion. The screened Coulomb interaction is represented by the energy cost U of double occupation of a single orbital. Without the Coulomb repulsion, the Hamiltonian is diagonal in momentum space, while with only the Coulomb interaction it is diagonal in real space. When U is larger than the kinetic energy bandwidth, we are in the strong coupling regime, whereas we are in the weak coupling regime when the kinetic energy dominates. In practice, the cuprates are very near intermediate coupling where neither one or the other term clearly dominates. This means that the exact low energy eigenstates have components in a sizeable fraction of the exponentially large Hilbert space, by contrast with the weakly interacting case, for example, where one can simply fill a Fermi sea and include perturbative corrections. Exact solutions to this model exist only in one-dimension or in infinite dimension. The two-dimensional case is the relevant one for cuprates. WEAK COUPLING Early progress towards explaining how superconductivity can arise from purely repulsive interactions came from weak coupling approaches. As explained by Paramekanti, the first phenomenon that comes out simply from the Hubbard model is the tendency towards antiferromagnetism. In regions where the system is not antiferromagnetic, spin fluctuations can nevertheless be present. The antiferromagnetic spin waves can then play the role of phonons: the glue that binds Cooper pairs. One important prediction of this approach [3-5], made first in the context of organic and heavy fermions superconductors in Ref. [3] by a group that included Claude Bourbonnais in Sherbrooke, is that the order parameter will have d-wave symmetry. The d-wave A-M.S. Tremblay <Andre-Marie. Tremblay@ USherbrooke.ca>, Département de physique et RQMP, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1 et membre du programme Matériaux quantiques, Institut canadien de recherches avancées, Toronto, ON, M5G 1Z8 LA PHYSIQUE AU CANADA / Vol. 67, No. 2 ( avr. à juin 2011 ) C 105 HIGH-TEMPERATURE SUPERCONDUCTIVITY ... (TREMBLAY) symmetry was also predicted for the cuprates [6,7], where it is indeed observed. with dynamical mean-field theory, the subject of our next section. Weak coupling ideas can serve as a stepping stone to develop non-perturbative approaches. The Two-Particle-Self-consistent (TPSC) approach does not use Feynman diagrams [8,9]. Instead, it enforces conservation laws, the Pauli principle and makes one ansatz that allows double occupancy to be determined selfconsistently. The validity of this approach as an approximate solution to the two-dimensional Hubbard model can be tested by comparing with Quantum Monte Carlo calculations [8-10]. The great advantage of Monte Carlo is that it is exact within statistical uncertainties that can be estimated. The disadvantage is that since fermions are in antisymmetric states, a sign (±) has to be attached to each Monte Carlo configuration. That prevents the sampling to be efficient at low temperatures in many cases. Nevertheless, there are some non-trivial regimes where Monte Carlo can be used as a benchmark for TPSC [10,11]. STRONG COUPLING Within TPSC one other important theorem is satisfied: The Mermin-Wagner theorem states that you cannot break a continuous symmetry in two dimensions. Hence, the phase transition that would appear in mean-field theory is replaced by a crossover temperature below which the antiferromagnetic correlation length ξ grows rapidly. When it becomes larger than the thermal de Broglie wavelength, ξth, that you can find from Δε ∼ υFΔk ~ υFξth-1 ~ kBT, then the electrons see the background spin configuration as if it were almost ordered. That leads to the opening of a pseudogap in the parts of the Fermi surface that would become gapped if there were longrange order [9,12]. Neutron scattering has determined that, in electron-doped cuprates, the pseudogap indeed opens up when the condition ξ ∼ ξth is satisfied [13]. The Fermi velocity is obtained from photoemission experiments, which are also in excellent agreement with TPSC [14]. In this approach, one finds a superconducting dome and the most recent calculations of conductivity are promising [15]. In particular, experiment finds, at the antiferromagnetic critical point and at larger doping, that there is a correlation between Tc and the size of the linear term in the resistivity [16,17]. The reason this approach works for electron-doped systems, even though it is based on weak coupling ideas extended to a non-perturbative regime, is that for electron doping, the interaction strength U is about 6t (where 8t is the bandwidth) which is below the value necessary to drive a strong Mott transition. The system can still be an insulator at half-filling above the Néel temperature because of fluctuating antiferromagnetism in two dimensions [9]. Or, it is also possible that U changes a bit with doping. Comparison of cluster perturbation theory calculations with experimental photoemission results have provided insight on the Fermi arc structure observed in the pseudogap regime and suggested that U is a bit smaller for electron-doped cuprates as compared with hole-doped [18]. This was confirmed by more recent calculations [19] based on first-principles methods combined 106 C PHYSICS IN CANADA / VOL. 67, NO. 2 ( APR.-June 2011 ) The solution of the Hubbard model in infinite dimension was a breakthrough that led to a new approach to the problem at strong coupling. Georges and Kotliar [20] as well as Jarrell [21] proposed that one could study the Hubbard model using an approasch that has the same structure as the solution in infinite dimension, namely a single interacting site in a self-consistent bath of non-interacting electrons. The problem with the singlesite approach [22] is that high-temperature superconductors are quasi two-dimensional and that actual observations of electronic properties by photoemission have shown that the self-energy is also momentum dependent. Hence, one cannot completely neglect spatial correlations. A generalization was thus proposed wherein a cluster of sites (instead of a single site) is embedded in an infinite bath of non-interacting electrons. There are two popular ways of doing this [23,24] that become identical if the cluster is large enough. Generically one calls these methods quantum cluster methods [25,26]. Some of the main results that have been found are as follows. a) Finite-size studies confirm that there is a d-wave superconducting transition at finite temperature [27,28]. b) The zero-temperature phase diagram contains antiferromagnetism and d-wave superconductivity with a tendency to coexist in the underdoped regime [29,30]. Such a tendency is observed, but not to the extent suggested by the calculations. It is difficult to study the stripe phase with these methods because of the large clusters that would be necessary, so one cannot rule out that stripe order would appear in this model, as it does in some experiments. c) Calculated spectral weights in the normal state are similar to those observed experimentally [18,30]. d) Retardation effects that have been associated with the glue in ordinary superconductors are also observed in the quantum cluster calculations. The relevant frequencies are associated with spin fluctuations, providing direct evidence for that mechanism [31,32]. e) In the presence of a small orthorhombicity, very large anisotropy in the transport coefficients is obtained if U is larger than the critical U for the Mott transition [33,34]. f) If one does not allow long-range order and lets the normal state survive at arbitrarily low temperature, one predicts that if the interaction strength U is larger than the one needed for the Mott transition, there appears a first-order transition between two metallic phases with different compressibilities and spin fluctuations [35,36]. That first-order transition ends at a critical point at extremely low temperature when U is large. At temperatures above that critical point, the scattering rate is large. This reminds us of the phenomenology associated with the quantum critical point discussed in the paper by Louis Taillefer in this issue, except that, this time, long correlation lengths are not necessary. HIGH-TEMPERATURE SUPERCONDUCTIVITY ... (TREMBLAY)AA WHAT HAVE WE LEARNED, AND SOME OF THE OPEN QUESTIONS Quantitative calculations that explain the many experiments are what is needed before the community agrees that the hightemperature superconductivity problem is solved. Progress in this direction has been steady and encouraging, as this contribution and that of Paramekanti have tried to show. One may need to refine the Hubbard model to explain every detail of the experiments, but what is encouraging up to now is that the hopping matrix elements suggested by band structure calculations and a value of U of the order of the bandwidth suffice to explain a lot of the phenomenology. It is noteworthy that the same methods reproduce the main features of the zerotemperature phase diagram of layered BEDT organics [37,38]. This phase diagram [37] and that of the high Tc suggest that antiferromagnetic fluctuations must be present for d-wave superconductivity but that frustration by doping or by exchange interaction must be large enough to prevent longrange antiferromagnetic order. The pseudogap in electron-doped cuprates seems well explained by precursors of antiferromagnetism [13,14], but the question is still open for hole-doped cuprates. Although holeand electron-doped cuprates must be modeled with interaction strengths U that are comparable, it could make a difference whether U is smaller or larger than the critical U for the Mott transition. In particular, a value of U larger than the critical U for the Mott transition allows one to explain spectral weight transfer as a function of doping, as observed in X-Ray absorption spectroscopy [39]. It would also provide a way to obtain large anisotropies (nematicity) in the pseudogap regime [34] and large scattering rates above optimal doping, but without the need for long magnetic correlation lengths [33]. It will be necessary to perform calculations that allow long-range antiferromagnetic order to find out whether the finite-doping transition between two metals [35,36] merges with the antiferromagnetic quantum critical point or is a different phenomenon. In the hole-doped cuprates, antiferromagnetism seems to end at a doping much smaller than the quantum critical point extrapolated from the pseudogap temperature. From a methodological point of view, including interactions between particle-hole pairs (so-called vertex corrections) in quantum cluster methods [40] would allow more reliable calculations of transport properties. We also have to face the fact that many experiments [41,42] suggest that to understand the whole phase diagram of hole-doped cuprates, it may be necesssary to include the effect of oxygens in the copperoxygen planes [43]. This means solving the three-band Hubbard model, a very difficult task that may stimulate further methodological improvements. The difficulty associated with strong correlations in hightemperature superconductors has forced us to accept that analytical methods cannot alone solve that problem. Much remains to be done but there is clear hope that the solution is in sight. And it is also clear that progress on that front has led and will continue to lead to many insights on interesting materials and perhaps even allow engineering heterostructures, as discussed in the contribution of Cavaglia, or new materials with properties we choose to design [44]. 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