# Charge-current correlation equalities for quantum systems far from equilibrium

### Submission summary

 As Contributors: Gunter Schütz Arxiv Link: https://arxiv.org/abs/1812.03020v2 Date submitted: 2019-02-07 Submitted by: Schütz, Gunter Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### Abstract

We prove that a recently derived correlation equality between conserved charges and their associated conserved currents for quantum systems far from equilibrium [O.A. Castro-Alvaredo et al., Phys. Rev. X \textbf{6}, 041065 (2016)], is valid under more general conditions than assumed so far. These correlation identities, which give rise to a current symmetry somewhat reminiscent of the Onsager relation, are also shown to hold in any space dimension and to imply a symmetry of the non-equilibrium linear response functions.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1812.03020v2 on 7 February 2019

## Reports on this Submission

Show/hide Reports view

### Strengths

- short, simple discussion of certain interesting symmetry relations
- generalised previous work to higher dimensions and find interesting consequences

### Weaknesses

- there is not a lot new that is actually done, relatively simple generalisations of known results
- claims of novelty are too strong, it is the same symmetry relation as that found in the literature
- citations to a fuller literature where such relations were found would be useful

### Report

In this paper, the author derive and discuss a particular symmetry relation for conserved currents. The symmetry relation is a consequence of space-time stationarity and the conservation laws, as well as some conditions at infinity. The relation is derived in arbitrary dimension, and some interesting consequences on response functions are discussed.

This is a very short paper, concentrating on a specific relation. The discussion is interesting, putting together some bits and pieces found in the literature, with a slight generalisation to higher dimensions and some of their consequences. But adjustments are necessary before it can be published.

First, it is important to mention that the relation (1) (and (41)) was found before, in one dimension, with the same proof. I think the relation is *not* here established in the more general not-necessarily-commuting setting than in [1,2] or other papers before, see below. Maybe also already mention that the higher-dimensional version is obtained by projecting to one dimension.

Second, as emphasised in the derivations in literature, it should be made more clear what the assumptions are from the outset. On the first page the phrase "stationary invariant quantum system", and then later "translation invariance", is not too clear. What is required is that the *state* be space-time stationary, and the conservation laws hold, along with appropriate asymptotic conditions on correlation functions.

Third, I would also ask the authors to reduce the claims of surprise and novelty; for instance after equations 24,25, or the sentence "This result is remarkable as there is no a priori reason to expect any such general equality for correlations of physically unrelated conserved quantities and their currents." These are subjective, and surely for many people, these are not remarkable, consequences of known sum rules and mostly written in the literature already. I also have the feeling the way the paper is written sometimes "diminishes" the work done previously. For instance, the correlation equality in [4] is not a "specific correlation equality", it is the same equality as (1) (in one dimension); and I'm not sure why use the phrase "phenomenological hydrodynamics perspective" for the derivation in [4], as the same projection principles as those of [5] and of the Mazur bound are used (in all cases, I don't think "phenomenological" is the appropriate term).

Here are more explanations:

The relation (1), and its consequence (41), are relatively well known at least in some community. The authors cite [1] where a similar relation is derived in one dimension assuming commutativity of the stationary density matrix with the charges. They also cite [2] where it is derived in one dimension without this assumption (although [2] may be seen as almost simultaneous with the present paper). In both cases quantum systems were in mind. But the relation is also reviewed in [eq 2.30, H. Spohn, Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains], there in the case of three conserved currents but the statement makes it clear it is general. It is mentioned that the proof uses only space-time stationarity and the conservation laws, found in [H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains (2014)]. In these cases, classical systems were discussed, but the proof makes it clear that it does not depend on classicality. In fact, this type of relation, following from conservation laws and sum rules, have been studied for a long time, see for instance the book "Hydrodynamic fluctuations, broken symmetry, and correlation functions" (1975) by Dieter Forster.

The authors make the claim that the relation (1) they find, without using commutativity, is derived in [1] using commutativity, and it is different from that derived in [2] without commutativity because of the ordering of the operators on the right-hand side. They therefore claim that they have a relation derived in a more general setting in the quantum context. I believe this is incorrect. The relation is exactly the same as that from [2], no difference in operator order. See [eq B.5, 2], in both the left and right hand side, the indices keep the same order, exactly like in (1). It is in [1] that there is a different order of operators, which is reached by using commutativity: in [eq A1, 1], the first line of the derivation uses commutativity of the charge with the stationary density matrix. But skipping this first line, keeping the indices in the same order, one obtains (1). Hence, I would ask the authors to change this claim - in one dimension, it is the same relation as that found in the literature.

The derivation is also essentially the same as that in [1] or [2] or [H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains (2014)]: one uses space-time stationary and the conservation laws, in order to establish sum rules for quantities $D_i^{\alpha\beta}$. Hence also the derivation is not new, I think it would be useful for the readers if the authors could clarify this.

What is different from previous works is that here the relation is derived in higher dimensions. This is done by projecting to one dimension, integrating on the transverse components. Basically, once this projection is done, the derivation is the same as in one dimension.

Finally, small things:

- After eq 35: it's not $M$ and $Q_\alpha$ not assumed to commute (as there is no $M$ in (35)), it is $Q_{\alpha}$ and $Q_{\alpha'}$

- sum rules (24,25) are found in the literature, see [eq 2.21, 2] or [eq 2.29, H. Spohn, Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains].

### Requested changes

See the report:

1- mention that the relation (1) (and (41)) was found before, in one dimension, with the same proof.

2- make more clear what the assumptions are from the outset.

3- reduce the claims of surprise and novelty; try to better represent work done previsouly; keep to the objective facts.

4- the small things mentioned in the report.

• validity: good
• significance: ok
• originality: low
• clarity: high
• formatting: perfect
• grammar: perfect

### Strengths

- Non-trivial and general results

### Weaknesses

- General presentation a bit rushed

### Report

The authors show that the charge-current symmetry of equation 1 is valid on a generic density matrix (not necessarily commuting with all the conserved charges) and in any spacial dimension. The result of eq 1 was originally found in the context of Generalised Hydrodynamics but they show that it is actually more general. They give a fulfilling proof of their statement in any spacial dimension. The paper definitely deserves publication.

### Requested changes

-I would request the authors to give a broader introduction of their results. Namely they could specify how eq 1 was useful in Generalised Hydrodynamics and which physical implications it had for integrable systems.
- It seems to me that it should be made more clear that in Generalised Hydrodynamics eq 1 is a consequence of local equilibrium and therefore not a priori valid at short times. The authors instead claim that it is valid at any time, especially after some quench from an inhomogenous initial state. Is that correct? If yes it should be made more clear in the introduction.
-In the introduction they could mention what are the assumption for their proof. It seems to me that one request is that the state has sufficiently rapid decay of correlations, eq 28.
- below eq 34 they should define what is M and its physical meaning
- slightly above eq 20: it seems that the authors want to point out that the symmetry eq 41 implies no hydrodynamics shocks. This point deserve to be expanded and made a bit more clear, also by referring to Generalised Hydrodynamics theory.
- in the conclusion they authors says that the correlation equality may be used as a probe of of the underlying asymptotic GGE, but it is not clear how.
- in the conclusion the authors says that fluctuations are expected to be described by the KPZ class. It has been recently show [Phys. Rev. Lett. 121, 160603] that fluctuations are typically diffusive and in some cases super-diffusive, see Phys. Rev. Lett. 121, 230602 and arXiv:1812.02701. The authors may refer to these references maybe for their claim.

• validity: top
• significance: high
• originality: high
• clarity: good
• formatting: good
• grammar: perfect