Anomalous Luttinger equivalence between temperature and curved spacetime: From black holes to thermal quenches

This paper discusses nonlinear effects due to gravitational anomalies in 1+1d CFTs in a curved space, and the relationship with Luttinger's perspective on the relation between temperature gradients and gravity. My honest sense is that this paper is in some sense a commentary on well-established ideas about thermal CFTs in a curved background, and remarking on the breakdown of the calculation from Luttinger's linear response calculation? That could be valuable to the literature.

I think my main concern - or at least confusion - about this manuscript is the following. A notion of local temperature is probably only well-defined in curved spacetime with slowly varying curvature. Equilibrium is sharply defined: I can ask what is the profile of <T^mu nu> given a classical gravitational background, and the anomalous energy-momentum equation subject to boundary conditions. Assuming no quantum gravity, I can exactly solve those equations without ambiguity. But then, it is far from clear to me whether particular experimental probes of temperature correspond to what the authors think they do? Some possibly semantic questions about whether some particular definition of temperature need to be corrected at the nonlinear level, might not be that interesting to me personally, because I do not know whether the authors' definition of temperature is the one probed by any given experimental system. As one example, the discussion around Eq. 56 and 57 to me seems very possibly compatible with this idea: I start with some sharply varying temperature profile in flat space and, depending on interpretation, may assign it different notions of "local temperature"?

What an experiment could clearly (in principle) measure are the correlation functions of <T^mu nu>, which the authors calculate. But then, this calculation seems to be based on an assumption of how an external temperature profile would couple to the theory, and again, I am not sure how confident I am in that theory on arbitrarily short length scales. Maybe I am the one who does not understand.

I am confused around Eq. (53). Are the authors assuming that the temperature gradient is constant while the energy current is not? That seems to require a very special type of heat bath. It could just as easily be that the actual temperature varies in some nonlinear way, so that the energy current is better behaved. I do not see how this discussion can feasibly relate to most experimental systems, where if anything, it seems more plausible to argue some systems are approximately thermally isolated when electron-phonon coupling is weak. In this setting it is J_E that must be constant, while T(x) could vary quite nonlinearly. If the system is not thermally isolated, then the thermodynamics of the phonon bath must also be accounted for?

In Section 5.2, the authors have a discussion of dynamics after a quench. From my perspective this sounds like a very straightforward problem where gravity is not needed at all: there is a T_L and T_R associated to the left and right movers of the CFT, and the profiles of T_L and T_R simply propagate at velocity c. If this is all the authors are commenting on from the gravitational perspective, it would be helpful to understand the role that the anomaly is playing more carefully by adding some concrete equations showing the imprint of the anomaly after the spacetime becomes flat? For example, if I encode some initial inhomogeneity in energy density/energy current via coupling to background spacetime, and then quickly shut off this coupling and return to flat space, presumably once the coupling is shut off the anomalous terms disappear from the energy momentum equation. Then, is there any anomalous part to the energy-momentum tensor after the quench? I fail to see how it is possible, once the quench ends the spacetime would be flat? Perhaps again, the only way the anomaly is encoded is in the precise way the theory is coupled to the external heat bath, and it's far from clear to me that there is only one logical way to couple to this heat bath, especially if the proposed temperature varies on extremely short length scales.

I think this commentary should be addressed or refuted in a revised manuscript, but I would probably encourage publication afterwards.

The manuscript discusses corrections to Luttinger relations due to quantum anomalies. It is clearly written, and the computations are easy to follow. In my opinion the only aspect of the manuscript that needs to be expanded is the discussion that the particles transport ballistically, and the two species don’t thermalize. This assumption for thermal systems seems to be very strong and in general valid for temperatures close to zero. Therefore, I would expect that it breaks down quite quickly as the temperature rises. However, I have not seen any discussion of the applicability of the formulas presented in the paper. Moreover, the validity of the corrections derived in this work requires that different species of fermions are kept at different temperatures. Again, no discussion of how close to realistic systems is this requirement, is included. I do not expect these relations to hold near black holes.