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Higherform symmetries, anomalous magnetohydrodynamics, and holography
by Arpit Das, Ruth Gregory, Nabil Iqbal
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Authors (as registered SciPost users):  Arpit Das · Nabil Iqbal 
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Preprint Link:  https://arxiv.org/abs/2205.03619v2 (pdf) 
Date submitted:  20220816 08:57 
Submitted by:  Das, Arpit 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Abstract
In $U(1)$ Abelian gauge theory coupled to fermions, the nonconservation of the axial current due to the chiral anomaly is given by a dynamical operator $F_{\mu\nu} \tilde{F}^{\mu\nu}$ constructed from the fieldstrength tensor. We attempt to describe this physics in a universal manner by casting this operator in terms of the 2form current for the 1form symmetry associated with magnetic flux conservation. We construct a holographic dual with this symmetry breaking pattern and study some aspects of finite temperature anomalous magnetohydrodynamics. We explicitly calculate the charge susceptibility and the axial charge relaxation rate as a function of temperature and magnetic field and compare to recent lattice results. At small magnetic fields we find agreement with elementary hydrodynamics weakly coupled to an electrodynamic sector, but we find deviations at larger fields.
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Reports on this Submission
Anonymous Report 3 on 2022914 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.03619v2, delivered 20220914, doi: 10.21468/SciPost.Report.5693
Report
This work studies magnetohydrodynamics in the presence of an AdlerBellJackiw anomaly from the perspective of higherform symmetries. The latter are in general suitable for describing the hydrodynamics of dynamical U(1) gauge fields; however, the ABJ anomaly presents us with challenges when trying to describe these systems in terms of symmetry principles, as the anomaly introduces relaxation terms as well as instabilities.
Due to these reasons, this paper is timely and has potential insights for future research on this topic. I therefore recommend the paper for publication, after the following comments/questions have been addressed:
One important result in the manuscript is the evaluation of the axial charge relaxation constant $\Gamma_A$. As discussed in the conclusion, ref. [2] recently found an inconsistent answer when estimating this constant using MHD, while the present holographic calculation shows consistency using the same method. The authors mention that their holographic result may then suggest that MHD, at small magnetic field, should be sufficient to compute $\Gamma_A$. However, it's not clear to me that the analysis presented supports this conclusion. First, 1/N corrections, which are of course suppressed in the holographic calculation, might potentially affect the evaluation of $\Gamma_A$ (these may not necessarily be captured by MHD loops). Second, it would be good to comment on whether (3.2) is the most general action consistent with various requirements discussed around that equation. Could there be additional couplings that might lead to a change in $\Gamma_A$ without affecting the axial anomaly?
Additionally, I found a few typos:
Eq. (3.17) indicates that $S_c$ has "plus" sign, which seems opposite to (3.18).
P. 12, last line: $F$ > $F_2$
P. 26, first line: "the the", secondlast line: "we we"
Report 2 by Luca Delacrétaz on 2022912 (Invited Report)
 Cite as: Luca Delacrétaz, Report on arXiv:2205.03619v2, delivered 20220912, doi: 10.21468/SciPost.Report.5679
Report
This manuscript explores the hydrodynamic regime of QFTs that exhibit an AdlerBellJackiw (ABJ) anomaly. Unlike 't Hooft anomalies for global symmetries, which are formulated nonperturbatively and have clear consequences on current algebras, nonperturbative consequences of a 4d ABJ anomaly are less clear. A dogmatic nonperturbative hydrodynamic approach would dismiss the chiral charge, as it is not conserved. The authors argue however that this would miss important physics.
The authors study such systems using a holographic model, in the probe limit. They also present an interesting comparison to a classical lattice simulation of a system with the ABJ anomaly, Ref. [2].
I recommend this manuscript for publication, but have several questions that the authors might wish to address:
 Could the authors be more explicit about the "important physics" that would be missed by dismissing the axial charge? Presumably what is meant is that axial charge may be parametrically longlived despite being nonconserved, even at strong coupling in the presence of an ABJ anomaly. And do they believe that this is a consequence of the ABJ anomaly alone, or together with an additional assumption such as weak coupling or holography?
 Could they briefly comment on whether they expect these conclusions to hold in light of the recent nonperturbative formulation of an ABJ anomaly, Refs. [22, 23]? The topological operators found in those papers carry a discrete label, suggesting that they might not carry consequences for hydrodynamics.
 I have a question concerning Sec. II. In this section, the authors review an calculation from Ref. [1] that produces a relaxation rate for the axial charge $\Gamma_A\propto B^2$. However, could the axial charge relax in the absence of a magnetic field? One way to compute its relaxation rate perturbatively in the anomaly $k$ is from its Kubo formula
$$\Gamma_A
= \frac{1}{\chi} \lim_{\omega\to 0} \frac{1}{\omega} {\rm Im} G^R_{\partial_0j_A^0\partial_0j_A^0}(\omega,q=0)$$
One can reproduce the authors' result (2.7) by linearizing the ABJ anomaly around a background field $B$, giving $\partial_\mu j^\mu_A\simeq k B\cdot \delta E$ (up to factors), leading to
$$
\Gamma_A \simeq \frac{(kB)^2}{\chi} \lim_{\omega\to 0} \frac{1}{\omega} {\rm Im} G^R_{\delta E \delta E}(\omega,q=0)
\simeq \frac{(kB)^2}{\chi }\frac{1}{\sigma}\, .
$$
Now one might expect that the Kubo formula produces a nonzero answer even when the background field vanishes. It could be evaluated within this perturbative framework through a hydrodynamic loop. While this contribution may be suppressed in the holographic model considered by the authors, it may be appreciable "in real life", at least at small fields.
 A minor point on Eq. (4.3): susceptibilities are usually defined as $\chi = d\langle j^0\rangle / d\mu$ rather than $\chi = \langle j^0\rangle / \mu$.
 Around Eq. (6.15), an improved comparison with MHD is made by accounting for the field dependence of the susceptibility. Can the field dependence of the conductivity be ignored here?
Author: Arpit Das on 20221116 [id 3026]
(in reply to Report 2 by Luca Delacrétaz on 20220912)
Q. Could the authors be more explicit about the "important physics" that would be missed by dismissing the axial charge? Presumably what is meant is that axial charge may be parametrically longlived despite being nonconserved, even at strong coupling in the presence of an ABJ anomaly. And do they believe that this is a consequence of the ABJ anomaly alone, or together with an additional assumption such as weak coupling or holography?
A. There are (at least) two answers to this: one is that this analysis appears to show that at small magnetic fields the decay rate vanishes as B^2 (where B is the magnetic field). If this is correct, it can indeed be made parametrically small, and one might seek a universal description that works at least in this regime. We currently believe that the above statement does not need any further assumptions like weak coupling or holography, however, we do not yet have a fully universal description and so this remains a conjecture. (We do note however that each factor appearing in the expression for this decay rate (2.7) can be given a fully universal meaning, so we believe this is a wellmotivated conjecture; though see below).
A further note is that the dynamics of the 2form current have a universal hydrodynamic sector, and the dynamics of the 1form axial current are clearly correlated with this through the anomaly equation, so at least some aspect of the axial current should admit a universal description.
Q. Could they briefly comment on whether they expect these conclusions to hold in light of the recent nonperturbative formulation of an ABJ anomaly, Refs. [22, 23]? The topological operators found in those papers carry a discrete label, suggesting that they might not carry consequences for hydrodynamics.
A. We believe this new nonpertubative formulation is extremely interesting and overlaps nicely with the philosophy of our work by explaining exactly which universality class we are working in. It’s important to note that the topological operators in those works aren’t really discrete: they are labelled by rational numbers, and now one can get into philosophical worries about whether the rationals are continuous or not. We believe a useful operational way to think about it is that the almostcontinuous noninvertible symmetry requires a protected local operator relation relating the nonconservation of the axial current to the topological density, and in this sense is maybe more like a continuous symmetry than a discrete one.
Indeed we are currently trying to use this insight to write down a universal hydrodynamic effective theory that implements this new symmetry. (Though the work is ongoing, we do seem to find that consistency with the noninvertible symmetry does lead to nontrivial constraints that reproduce some of the chiral MHD phenomenology).
Q. ...Now one might expect that the Kubo formula produces a nonzero answer even when the background field vanishes. It could be evaluated within this perturbative framework through a hydrodynamic loop.
A. This is a good point; the result \Gamma_A \sim B^2, suggests that the relaxation rate vanishes in the limit of vanishing magnetic field. It is, at the moment, not clear to us whether this is an artifact of the classical description. We like the referee’s formulation in terms of a Kubo formula and agree with the referee that it is possible that when one includes fluctuations there is a nonvanishing relaxation rate even at zero magnetic field. It also seems possible that the appropriate lowfrequency correlator always vanishes due to special properties of the topological density (as appears to happen in the classical limit we study). We think that answering this properly is beyond the scope of this paper, but we leave this for further investigation, and we thank the referee for this comment.
Q. A minor point on Eq. (4.3): susceptibilities are usually defined as \chi=\frac{d<j_A>}{d\mu_A} rather than \chi=\frac{<j_A>}{\mu_A}
A. We thank the referee again for correctly pointing this out and we have made the suggested changes regarding the definition of susceptibility.
Q. Around Eq. (6.15), an improved comparison with MHD is made by accounting for the field dependence of the susceptibility. Can the field dependence of the conductivity be ignored here?
A. In the holographic model in the probe limit, it actually seems that the resistivity (i.e. the inverse conductivity) is independent of the background field, as the membrane paradigm fixes it to be equal to the quantity \Sigma(r) in (5.3). Thus the comparison that we perform (i.e. taking into account the field dependence of the susceptibility) seems like the only one that we could do.
Author: Arpit Das on 20221116 [id 3027]
(in reply to Report 3 on 20220914)Q. First, 1/N corrections, which are of course suppressed in the holographic calculation, might potentially affect the evaluation of \Gamma_A (these may not necessarily be captured by MHD loops)
A. In this work, we have neglected the backreaction of the charge degrees of freedom on the geometry. Our calculation is also entirely classical, in that we have ignored fluctuations, which in this framework are suppressed by 1/N with N a proxy for the number of fieldtheoretical degrees of freedom. It is reasonable to ask whether such effects will change the picture above. As the actual lowfrequency calculation in the bulk essentially exactly parallels the hydrodynamic calculation (given in Sec. 2), it seems reasonable to expect that such corrections would change individually the values of quantities such as \Gamma_A and the resistivity \rho but not change the relationship between them that we find in Eq.(5.12). (This is broadly the expectation from the usual fluidgravity correspondence.)
Q. Second, it would be good to comment on whether (3.2) is the most general action consistent with various requirements discussed around that equation. Could there be additional couplings that might lead to a change in \Gamma_A without affecting the axial anomaly?
A. In the action given in Eq.(3.2), one could have (e.g.) higher derivative terms consistent with the symmetry structure described above; we believe that they would have effects qualitatively similar to those in the point above, in that they would not affect the functional form of relationships such as (5.12) though they might change individual quantities such as \rho, etc. At larger B fields, however, it seems likely that the functional form of the answer (i.e. computed numerically in Figure 2) is not protected and likely depends on all couplings.
Q. Additionally, I found a few typos...
A. We thank the referee for pointing out the typos which have been now fixed.