# Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

### Submission summary

 As Contributors: Denis Bernard Arxiv Link: http://arxiv.org/abs/1706.03984v3 Date submitted: 2017-09-07 Submitted by: Bernard, Denis Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Condensed Matter Physics - Theory

### Abstract

Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.

#### Current status:

Has been resubmitted

1- As suggested by the referee, we modified the introduction, which is now shorter and hopefully clearer and less technical. To satisfy the referee's demand, we have suppressed some of the equations present in the introduction. But we believe that there is some usefulness in presenting the structure of the equations governing the phenomena we aim at describing, even-though some of the terms entering those equations are loosely defined at this stage (they are fully defined in the text).

2- The term « stochastic Lindblad equation » was made in analogy with the already existing name « stochastic Schroedinger equation » which codes for some stochastic deformation of the Schrodinger equation. The equations we deal with are stochastic extensions of the Lindblad equation (which is a deterministic, non-random, evolution equation). There is not a unique way to consistently extend stochastically a Lindblad equation given its deterministic part. For instance, quantum trajectory equations and our equations are both stochastic equations and have identical deterministic drift parts. Thus we had to introduce a new name to differentiate both types of equations which describe different physical processes. Thanks to the referee comment, we understand that the name we choose did not reach our aim, so we remove it except at one instance.

3- We are surprised by this comment (for instance, one would not ask this question in the case of classical stochastic equations of the Langevin type). We are dealing with the stochastic differential equation because it codes for the fluctuations of all observable quantities. The Lindblad equation, which is the deterministic part of the equation, only codes for the evolution of the mean (over the noise) of those quantities (in quantum theory there are two origins of randomness: that due to the noise and that due to the probablistic nature of quantum mechanics). This is standard in probability theory and does not require further explanation. Let us also point out that it is in general not true that the Lindblad generator completely fixes the form of the noise term of the density matrix evolution equation: to fix this term given the drift term one has to invoque extra information, say specifying which physical processes are at play (e.g. monitoring or unitary interaction with extra degrees of freedom), or specifying the nature of the noise (for instance, the noise can be Brownian or Poisson like). Moreover, unravelling the deterministic Lindblad equation in a stochastic differential equation (classical or quantum) offers some technical advantages (for instance in identifying the mean dynamics of the slow mode in the XXZ case).

4- Sorry. We corrected it.

5- We suppressed the Appendix G which contained extra information not explicitly used in the main text.

6- We corrected the misprints we found in the text.

### Submission & Refereeing History

Resubmission 1706.03984v4 (13 October 2017)
Resubmission 1706.03984v3 (7 September 2017)
Submission 1706.03984v2 (23 June 2017)

## Invited Reports on this Submission

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### Anonymous Report 1 on 2017-9-25

• Cite as: Anonymous, Report on arXiv:1706.03984v3, delivered 2017-09-25, doi: 10.21468/SciPost.Report.245

### Report

1)
The authors have improved the introduction by removing some of
the technical parts. In my opinion it has become more reader-friendly
after the revision.

3)
Despite the authors’ surprise, I believe that the question
has its right of existence. Indeed, the Langevin formalism is
by far much better founded and understood for classical than
for quantum systems. From Eq. (4) of the manuscript it does
not become clear to me, what is the freedom of choice (after
fixing the $e_j$ Lindblad operators) in the stochastic version
of the equation? What extra assumptions (not present in
system-bath coupling and how do they yield different
Langevin-type equations? In their reply, the authors have
already given some keywords. However I would find it rather
important to include a short discussion about the construction
of quantum Langevin equations in the manuscript, since right
now there is very little information given, despite being
an object of central importance.

On the other hand, it is not true that the Lindblad formalism
only codes for mean quantities. In fact, there have been
successful attempts to calculate even large deviation functions
for spin chains (see e.g. PRL 112, 040602 (2014)) without
invoking the stochastic calculus method.

### Requested changes

Include a better discussion of quantum Langevin equation

• validity: high
• significance: good
• originality: high
• clarity: good
• formatting: good
• grammar: excellent