We show that, at weak to intermediate coupling, antiferromagnetic fluctuations enhance d-wave pairing correlations until, as one moves closer to half-filling, the antiferromagnetically-induced pseudogap begins to suppress the tendency to superconductivity. The accuracy of our approach is gauged by detailed comparisons with Quantum Monte Carlo simulations. The negative pressure dependence of Tc and the existence of photoemission hot spots in electron-doped cuprate superconductors find their natural explanation within this approach.

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Carbon budgets, which define the total allowable CO2 emissions associated with a given global climate target, are a useful way of framing the climate mitigation challenge. In this paper, we review the geophysical basis for the idea of a carbon budget, showing how this concept emerges from a linear climate response to cumulative CO2 emissions. We then discuss the difference between a CO2-only carbon budget associated with a given level of CO2-induced warming, and an effective carbon budget associated with a given level of warming caused by all human emissions. We present estimates for the CO2-only and effective carbon budgets for 1.5 and 2 °C, based on both model simulations and updated observational data. Finally, we discuss the key contributors to uncertainty in carbon budget estimates, and suggest some implications of this uncertainty for decisionmaking. Based on the analysis presented here, we argue that while the CO2-only carbon budget is a robust upper bound on allowable emissions for a given climate target, the size of the effective carbon budget is dependent on the how quickly we are able to mitigate non-CO2 greenhouse gas and aerosols emissions. This suggests that climate mitigation efforts could benefit from being responsive to a changing effective carbon budget over time, as well as to potential new information that could narrow uncertainty associated with the climate response to CO2 emissions.

While we agree with the numerical results of Ref.[1], we arrive at different conclusions: The apparent opening of a gap at finite temperature in the two-dimensional weak-coupling Hubbard model at half filling does not necessitate an infinite correlation length (Slater mechanism) nor a thermodynamic finite-temperature metal insulator transition (MIT). The pseudogap is a crossover phenomenon due to critical fluctuations in two dimensions, namely, to the effect of a ; spin-density wave (SDW) that is large compared with the thermal length.We use the units of Ref.[1]. The inset of Fig. 1 shows hn " n # i obtained in Ref. [1] for N c 36, U 1 and N c 64, U 0:5 along with the corresponding results obtained [2] from the local moment sum rule T=N c P q sp q 1 ÿ 2hn " n # i, supplemented with the relations ÿ1 sp q 0 q ÿ1 ÿ U sp 2 and U sp Uhn " n # i= hn " ihn # i. Figure 1 also shows the pseudogap in the density of states ! obtained from [3] s k Un ÿ U 8T N P q 3U sp sp q U ch ch qG 0 k q which includes the effects of both spin sp and charge ch fluctuations and satisfies 1 2 Tr s G 0 Uhn " n # i. The charge fluctuations are constrained by the sum rule T=N c P q sp q ch q 1. As temperature is lowered from T 1=20 to 1=22 and 1=32, the pseudogap in ! quickly deepens. The distance between the two peaks is in quantitative agreement with Ref.[1]. In addition, extensive comparisons with quantum Monte Carlo (QMC) have shown earlier [2,3] that our approach agrees quantitatively with QMC, and contains the same finitesize effects. In particular, ! 0 is smaller in smaller lattices. Hence, while at T 1=32 the criterion [1] ! 0 < 1 10 ÿ2 is satisfied for N c 64 and short ' N 1=2 c , we still need to verify that this reflects the behavior of a large (but not infinite) correlation length in the thermodynamic limit. That is why we verified that ! 0 < 1 10 ÿ2 for N c 128 2 as well. ! 0 in dynamical cluster approximation (DCA) has the opposite size dependence and satisfies ! 0 < 1 10 ÿ2 for N c 64. Note that for size 128 2 , already reaches 40 lattice spacings at T 1=22. All of the above results may be understood analytically from the above equations [2] by considering the limiting case where the characteristic frequency in the spin spectral weight 00 sp becomes smaller than temperature (renormalized classical regime). The local moment sum rule prevents a finite-temperature mean-field transition by letting U sp , and hence hn " n # i, exhibit a downturn at T . Below that temperature, grows rapidly but it becomes infinite only at T 0. Similarly, the opening of the pseudogap with decreasing temperature can be traced [2], in d 2, to the singular contribution of sp q to s k when becomes larger than the single-particle thermal de Broglie wavelength th v F =T. Indeed, in that limit, the singleparticle spectral weight Ak H ; ! at hot spots is given by ÿ2 00 ! ÿ 0 2 002 ÿ1 with ! ÿ 0 0 and 00 k H ; 0 / 3ÿd = th . Since = th grows exponentially in the d 2 renormalized classical regime, Ak H ; ! can become exponentially small at ! 0 even without a ...