Stochastic dissipative quantum spin chains (I) :

Quantum fluctuating discrete hydrodynamics

Michel Bauer, Denis Bernard, Tony Jin

Institut de Physique Théorique de Saclay, CEA-Saclay CNRS, 91191 Gif-sur-Yvette, France, and Département de mathématiques et applications, ENS-Paris, 75005 Paris, France

Laboratoire de Physique Théorique de l’Ecole Normale Supérieure de Paris, CNRS, ENS PSL Research University, UPMC Sorbonne Universités, 75005 Paris, France.

*

March 12, 2021

## 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.

###### Contents

- 1 Introduction
- 2 Quantum stochastic dissipative spin chains
- 3 The stochastic quantum Heisenberg spin chain
- 4 Discussion and perspectives
- A Brownian transmutation
- B A spin one-half toy model at strong noise
- C Proof of the XXZ stochastic slow modes dynamics
- D Derivation of the slow mode mean dynamics
- E Strong noise limit and effective stochastic dynamics
- F Derivation of the XXZ mean diffusive equation
- G Fermionization of the quantum stochastic XY model

## 1 Introduction

Non-equilibrium dynamics, classical and quantum, is one of the main current focuses of both theoretical and experimental condensed matter physics. In the classical theory, important theoretical progresses were recently achieved by solving simple paradigmatic models, such as the exclusion processes [1, 2]. This collection of results culminated in the formulation of the macroscropic fluctuation theory (MFT) [3] which provides a framework to study, and to understand, a large class of out-of-equilibrium classical systems. In the quantum theory, recent progresses arose through studies of simple, often integrable, out-of-equilibrium systems [4, 5]. Those deal for instance with quantum quenches [6, 7, 8], with boundary driven integrable spin chains [9, 10], or with transport phenomena in critical one dimensional systems either from a conformal field theory perspective [11, 12, 13] or from a hydrodynamic point of view [14, 15]. However, these simple systems generally exhibit a ballistic behaviour while the MFT theory deal with locally diffusive systems satisfying Fick’s law. Therefore, to decipher what the quantum analogue of the macroscopic fluctuation theory could be –a framework that we may call the mesoscopic fluctuation theory–, we need, on the one hand, to quantize its set-up and, on the other hand, to add some degree of diffusiveness in the quantum systems under study.

The macroscopic fluctuation theory [3] provides rules for specifying current and density profile fluctuations in classical out-of-equilibrium systems. One of its formulation (in one dimension) starts from stochastic differential equations for the density and the current , the first one being a conservation law:

(1) | |||

with the diffusion coefficient, the conductivity and a space-time white noise, . Here is the size of the system, so that the strength of the noise gets smaller as the system size increases. The statistical distribution of the noise induces that of the density and of the current. The weakness of the noise for macroscopically large systems ensures that large deviation functions are computable through the solutions of extremization problems (which may nevertheless be difficult to solve). See refs.[3, 16, 17, 18] for instance.

The second equation in (1) is a constraint, expressing the current in terms of the density plus noise. A direct quantization of the evolution equations (1) seems difficult because of their diffusive nature and because, in a quantum theory, a constraint should be promoted to an operator identity. However, we can choose to upraise these two equations into dynamical equations, of a dissipative nature, by adding current friction. For instance, we can lift these equations into the two following dynamical ones:

(2) | |||

where is again a space-time white noise. We have introduced a time scale to make these equations dimensionally correct and a dimensionless control parameter , so that the current friction coefficient is . In the large friction limit, , we recover the previous formulation. More precisely, let us rescale time by introducing a slow time variable and redefine accordingly the density and the current . By construction, these new slow fields satisfy the conservation law, , and the constraint , in the limit , (if ), with whose statistical distribution is identical to that of .

In other words, the slow modes and of the dissipative dynamical equations (2), parametrized by the slow times , satisfy the MFT equations (1), in the large friction limit. This is the strategy we are going to develop in the quantum case. Because equations (2) are first order differential equations (in time), they have a better chance to be quantizable. Quantizing these equations requires dealing with quantum noise. Fortunately, the notion of quantum stochastic differential equations exists and has been extensively developed in quantum optics [19] and in mathematics [20].

Quantum stochastic dissipative spin chains are obtained by coupling the quantum spin chain degrees of freedom to noise. The quantum evolution is then a random stochastic dissipative evolution. In the simplest case of the Heisenberg XXZ spin chain with random dephasing noise –the case we shall study in detail– the evolution is specified by a stochastic Lindblad equation of the following form

with the spin chain density matrix, the XXZ Heisenberg hamiltonian (whose definition is given below in eq.(17)) and the spin half Pauli matrix on site of the chain. These are stochastic equations driven by real Brownian motions , attached to each site of the spin chain. The dephasing noise induces friction on the spin current, in a way similar to the classical theory described above. The dimensionless parameter controls the strength of the noise: the bigger is , the stronger is the friction.

Motivated by the previous discussion, we look at the large friction limit . In this limit, dephasing occurs strongly and rapidly, and it induces strong decoherence and destructive interferences. As a consequence, only a subset of observables survives this limit: those which are invariant under spin rotations around the -axis, independently at each site, and under the action of the -rotation invariant part of the hamiltonian (recall that the XXZ hamiltonian decomposes into , with the so-called anisotropy: its invariant part is ). Projecting on these invariant observables yield a quantum analogue of Fick’s law, in a way similar to the classical theory we described above. Those observables possess a slow dynamics with respect to the slow time . As we proved below, the effective quantum dynamics is again described by a stochastic Lindblad equations of the following form

where are specific hoping operators from site to site dressed by the spin values at neighbour sites. See eq.(30,31) for their definitions. These are again stochastic equations driven by Brownian motions . There are three complex Brownian motions per link indexed by .

This effective dynamics codes for random incoherent hoping along the chain, with hoping amplitudes dressed by the neighbour spin values. This result has a simple interpretation: the XXZ hamiltonian generated coherent hoping from site to site, the on-site independent random dephasing induces decoherence so that phase memory is lost after any jump, and as a consequence, the coherent hoping process is transformed in an incoherent jump process. Here the incoherent jumps are from neighbour sites because we chose the on-site noise to be uncorrelated. Would we have chosen on-site noise with long distance correlations, the effective process would have included long distance hoping incoherent processes.

The effective slow dynamics can be written on observables (using the Heisenberg representation of Quantum Mechanics instead of the Schrödinger representation as above). For the spin observables, it reads

with noisy operators of a specific form, see eq.(32). It is clearly of a quantum, stochastic, diffusive, discrete hydrodynamic nature. In mean it codes for diffusion with constant diffusion constant, but it includes quantum, stochastic, fluctuations. It is worth comparing it with the classical MFT above.

Within a local hydrodynamic approximation –whose domain of validity remains to be more precisely specified– this quantum stochastic equation can be mapped into a classical, stochastic, discrete hydrodynamic equation whose formal continuous limit coincides with eq.(1). See eq.(36) below. In other words, within this approximation, classical MFT is an appropriate description of these quantum, stochastic, systems.

Notice that, when extracting the effective dynamics at large friction, we observe a Brownian transmutation –from real Brownian motions attached to the chain sites to complex Brownian motions attached to the links of the chain. In the original random dephasing XXZ model, there is one real Brownian noise per site, whereas in the slow effective dynamics there are three complex Brownian motions per link. This property is made mathematically precise in Appendix A.

These results are simpler in the XY model, which corresponds to the XXZ model with . Then there is only one complex Brownian motion per link and the jump process is simplified accordingly. See eq.(38).

This paper is organized as follows: In Section 2 we first define quantum stochastic versions of spin chains using tools from quantum noise theory. We then extract the relevant slow modes of those quantum stochastic systems and we describe their effective stochastic dynamics by taking the large friction limit of the previously defined quantum stochastic spin chain models. This general framework is illustrated in the case of the quantum stochastic Heisenberg XXZ spin chain in Section 3. In particular we describe how to take the large friction limit and how this limit leads to quantum fluctuating discrete hydrodynamic equations. A summary, extracting the main mechanism underlying this construction, as well as various perspectives, are presented in the concluding Section 4. We report most –if not all– detailed computations in seven Appendices from A to G.

## 2 Quantum stochastic dissipative spin chains

To quantize the dynamical MFT equations (2) we use quantum stochastic differential equations (SDEs) to couple a spin chain^{1}^{1}1Of course, we can formally extend this definition to higher dimensional lattices. to quantum noise. The quantum stochastic equations can be viewed as describing the interaction of a quantum system (in the present case, the spin chain) with infinite series of auxiliary quantum ancilla bits representing the quantum noise [21]. In our framework, there will be one quantum noise per lattice site in a way similar to the classical case in which there is one Brownian motion per position. The interaction between the spin chain and the quantum noise will be chosen appropriately to induce friction on the relevant spin currents.

### 2.1 Generalities

Quantum stochastic differential equations (SDE) define quantum flows of operators (in the Heisenberg picture) and density matrices (in the Schrödinger picture) on the tensor product of a system Hilbert space (in the present case, the spin chain Hilbert space) with the quantum noise Fock spaces. For operators , they are of the form

(3) |

with and the Hamiltonian and the (dual) Lindbladian of the quantum system dynamics, respectively. The operators ’s code for the noise-system coupling. The quantum noises are quantum operators , and their dual , acting on Fock spaces, with canonical commutation relations and quantum Itô rules^{2}^{2}2Here we restrict ourselves to diagonal Itô rules but the generalization to the non diagonal case is simple, see [19, 20, 22] for a brief introduction.

with the occupation numbers coding for the temperature of noise (the zero temperature case corresponds to ). The Lindbladian is of the form , where all Lindbladians and can be expressed in terms of the coupling operators . See e.g. [19, 20, 22] for more detailed information.

In the context of quantum stochastic spin chains, the index in labels the sites of the spin chain, the hamiltonian is that of the spin chain and the may come from an extra dissipative process acting on the spin chain in the absence of quantum noise. To specify the model we also have to declare how the spin chain degrees of freedom and the quantum noise are coupled by choosing the operators . By convention, these are operators acting locally on the site of the spin chain. (But this choice can of course be generalized to operators acting on neighbour spins). We shall describe explicitly the example of the quantum stochastic XXZ Heisenberg spin in the following Section 3.

To simplify the discussion we shall now assume that the ’s are hermitian. Then, up to a redefinition of the ’s, we may restrict ourselves to without loss of generality, so that the quantum stochastic differential equations reduce to stochastic differential equations (SDE). They read^{3}^{3}3We use Itô convention when writing stochastic differential equations (SDE).:

(4) |

where are classical Brownian motions normalized to . In this case, the derivatives and the Lindbladian are defined by and , respectively. The evolution equations for density matrices are the dual of eq.(4). They read:

(5) |

with and .

If furthermore , still with the ’s hermitian, the flow (4) is actually a stochastic unitary evolution with infinitesimal unitary evolutions with hamilonian generators

(6) |

with normalized Brownian motions. The stochastic evolution equation for the operator reads . The dual evolution equation for density matrices reads

(7) |

In this case, for each realization of the Brownian motions, the density matrix evolution is unitary, but its mean (w.r.t. to the Brownian motions) is dissipative (encoded in a completely positive map).

### 2.2 Effective stochastic dynamics on slow modes

The dimensionless parameter controls the strength of the noise and the mean dissipation. As we argued in the Introduction, we aim at taking the large friction limit in order to recover the quantum analogue of the macroscopic fluctuation theory (which we call the mesoscopic fluctuation theory).

The aim of this section is to describe a general enough step-up to deal with the large friction limit –which is also the strong noise limit– and determine the effective hydrodynamics of the slow modes in the limit of large dissipation . Since the aim is here to present a possible framework, we will not enter into a detailed description of any peculiar models but only present the general logical lines. A more detailed and precise description will be provided in the following Section dealing with the stochastic Heisenberg XXZ model.

We first have to identify what the slow modes are? In the limit , the noise induced dissipation is so strong that all states are projected into states insensible to these dissipative processes. There is a large family –actually an infinite dimensional family in the example of the Heisenberg spin chain below– of such invariant states. These are the slow modes. They are parametrized by some coordinates –actually an infinite number of coordinates. The effective hydrodynamics is the dynamical evolution of these coordinates parametrized by the slow time . (This is a slight abuse of language as we did not yet take the continuous space limit). In other words, the effective hydrodynamics is the dynamics induced on the slow mode manifold. It also describes the effective large time behaviour, which is dissipative and fluctuating by construction.

Let us first analyse the mean slow modes. Let be the mean density matrix, where the expectation is with respect to the Brownian motions . From eq.(5), it follows that its evolution equation is

(8) |

where we set . The maps and are operators, so-called super-operators, acting on density matrices. Since they are time independent, solutions of eq.(8) are of the form . Since and are non-positive operators (by definition of a Lindblad operator), with the projection operator on which is composed of states such that . In other word, , and this forms the mean slow mode set. This projection mechanism of states on some invariant sub-space is analogous to the mechanism of reservoir engineering [23].

Since the space of mean slow modes is of large dimension, there is a remaining slow evolution. It can be determined via a perturbative expansion to second order in , as explained in Appendix D. It is of diffusive nature and it is parametrized by the slow time . It reads (See the Appendix D for details, in particular we here assume that as otherwise we would have to redefine the slow mode variables to absorb the fast motion generated by .) :

(9) |

where is the super-operator, acting on density matrices in , via

(10) |

with the projector on and the inverse of the restriction of on the (orthogonal) complement of . Eq.(9) generates a diffusive flow on , the mean slow mode manifold, which is diffusive and dissipative even if the original spin chain dynamics was not (i.e. even if so that is purely Hamiltonian). The effective slow diffusion is generated by the on-site noise.

Since the evolution is stochastic there is more accessible information than the mean flow, and one may be willing to discuss the fluctuations and their large friction limit. A way to test this stochastic process is to look at expectations of any function of the density matrix. For instance, we may consider polynomial functions, say , and look at their means, say . This amounts to look for statistical correlations between operator expectations. Let be the expectations (w.r.t. the Brownian motions ) of those functions. As for any stochastic process generated by Brownian motions, their evolutions are governed by a Fokker-Planck like equation of the form

(11) |

with a second order differential operator (acting on functions of the random variable ). It decomposes into where is the Fokker-Planck operator associated to the noisy dynamics and is the first order differential operator associated to the deterministic dynamics generated by the Lindbladian .

Let us now identify what the slow mode observables are. Recall that these modes are those whose expectations are non trivial in the large friction limit . The formal solution of eq.(11) is

As a differential operator associated to a well-posed stochastic differential equation –that corresponding to the noisy part in eq.(4)–, the operator is non-positive. Hence the only observables which survive the large friction limit are the functions annihilated by , i.e. such that . The functions which are not in the kernel of have expectations which decrease exponentially fast in time with a time scale of order .

The slow mode observables are thus those in . Their evolution –in the limit at fixed – can again be found by a perturbation theory to second order in . See Appendix E for details. The same formal manipulation as for the mean flow, but now dealing with operators acting on functions of the density matrix, tells us that the effective hydrodynamic equations are of the form

(12) |

with be the projector on and the inverse of the restriction of on the complement of its kernel. Again, as above, we made the simplifying hypothesis that .

The above equation indirectly codes for the random flow on the slow modes. However, it may be not so easy, if not difficult, to make it explicit and tractable from this construction – although, it some case, such as in the XY model, it may be used to reconstruct the stochastic slow flow. So we now make a few extra hypothesis which will allow us to construct explicitly the stochastic flow on the slow modes.

Let us now suppose, that the initial stochastic dynamics is defined as in eq.(6) by the hamiltonian generator (i.e. we assume that and for all ). We furthermore assume that all local operators commute: . They generate commuting s actions. To simplify we furthermore assume that the hamiltonian has no s-invariant component (this hypothesis is easily relaxed and will be relaxed in the case of the XXZ model). Under this hypothesis the noisy dynamics, generated by , can be explicitly integrated. It is simply the random unitary transformation with . As a consequence, functions in , which, by definition, are invariant under such unitary flows, are functions invariant under all s generated by the operators . Hence, the slow mode observables are the functions invariant under all s,

(13) |

for any real ’s.

The hydrodynamic flow is thus a flow a such invariant functions. But functions over a given space invariant under a group action are functions on the coset of that space by that group action. Hence, the slow mode observables are the functions on the coset space obtained by quotienting the space of system density matrices by all s actions generated by the operators . Elements of this coset space are the fluctuating slow modes and the fluctuating slow hydrodynamic evolution takes place over this coset. These flows are defined up to gauge transformations. Indeed density matrices and , with some s transformations, represent the same elements of the coset space. If is the flow presented within the gauge , the flow in the gauge transformed presentation is with gauge transformed hamiltonian . See Appendix B for the discussion of the simple toy model of a spin one-half illustrating this discussion.

To explicitly determine the fluctuating effective dynamics we use the opportunity that the noisy dynamics can be exactly integrated to change picture and use the interaction representation. Let us define the transformed density matrix by

By construction, if is a -invariant function,

so that we do not lose any information on the stochastic slow mode flow by looking at the time evolution of the transformed density matrix (and this corresponds to a specific gauge choice). The latter is obtained from that of by going into the interaction representation via conjugacy, so that

(14) |

with

(15) |

Going to the interaction representation allows us to extract most – if not all – of the rapidly oscillating phases which were present in the original density matrix evolution. Theses phases were making obstructions to the large friction limit and their destructive interferences were forcing the expectations of non s-invariant functions to vanish. Once these phases have been removed, it simply remains to show that the evolution equation (14) has a well-defined limit as a stochastic process. This is described in details in the case of the XXZ model in the following Section 3.

### 2.3 Remarks

Let us end this Section with a few remarks.

— In the above discussion, we made a few hypothesis in order to simplify the presentation. Some of them can be relaxed without difficulties. First we supposed that , with the projector on , or that has no s-invariant component. This hypothesis can be relaxed, in which case one has to modify slightly either the perturbation theory used to defined the slow dynamics or the unitary transformation defining the interaction representation. This is actually what we will have to do in the case of the XXZ model – and this is one of the main difference between the XXZ and the XY models. Second, when discussing the change of picture to the interaction representation we assume that . This can also be easily removed. The only difference will then be that the evolution equations in the interaction representation are not going to be random unitaries but random completely positive maps. Finally, in order to implement the map to the interaction representation we assume that the operators ’s were commuting. This is actually necessary as otherwise we would not be able to integrate the noisy dynamics and thus we would not be able to implement the unitary transformation mapping to the interaction representation.

— The noisy interaction, coded by the coupling , can be viewed as representing the interaction of local degrees of freedom with some local reservoir. This interaction has a tendency to force the system to locally relax towards local states invariant under the noisy interaction. For instance if we choose the operators to be proportional to the local energy density these local invariant states are locally Gibbs. So the noisy interaction can be seen as enforcing some kind of local equilibrium or local thermalization if the noise-system is chosen appropriately. The typical relaxation time scale for these processes are proportional to so that the large limit then corresponds to very fast local equilibration. The slow mode dynamics can then be interpreted as some kind of a fluctuating effective quantum hydrodynamics. Here and in the following, we are making a slight abuse of terminology as, usually, hydrodynamics refers to the effective dynamics of slow modes of low wave lengths. We are here going to describe slow mode dynamics without taking the small wave length limit (i.e. the slow mode dynamics on a discrete lattice space). We refer to this limit as discrete hydrodynamics.

— The construction of the effective slow stochastic dynamics we are presenting relies on analysing the hydrodynamic limit at of the dynamics in the interaction representation. There, the hamiltonian generator is given by conjugating by the Brownian phase operator as made explicit in eq.(15). Implementing this conjugacy produces random phases of the following form (recall that )

(16) |

with real numbers. Here the equivalence relation refers to the fact that in law. These phases are random, irregular and highly fluctuating in the limit of large friction. Our proof of the effective slow dynamics relies on the fact that, surprisingly, when , these phases converge to complex Brownian motions. We refer to this property as “Brownian transmutation”. See Appendix A for a proof.

## 3 The stochastic quantum Heisenberg spin chain

We now illustrate the previous general framework in the simple, but non trivial, case of the XXZ Heisenberg spin chain. We first add noise to the usual Heisenberg spin chain model in a way to preserve the conservation law, as in the classical macroscopic fluctuation theory. We then describe the slow mode dynamics including its fluctuations and its stochasticity.

The XXZ local spins, at integer positions along the real line, are spin halves with Hilbert space . The XXZ Hamiltonian is a sum of local neighbour interactions, , with Hamiltonian density , so that^{4}^{4}4Here are the standard Pauli matrices, normalized to , with commutation relations and permutations. As usual, let . They satisfy and and .:

(17) |

where fixes the energy scale and is the so-called anisotropy parameter. We define and , so that . Let be the local density and be the local current. The equations of motion, , are:

with and . The first equation is a conservation law, the second codes for spin wave propagation. The simple case of the XY model, corresponding to , is described at the end of this Section.

### 3.1 The stochastic XXZ model

We now add noise and write the quantum stochastic equation in such way as to preserve the conservation law. This completely fixes the form of the quantum SDE. Indeed, demanding that the conservation law holds in presence of quantum noises imposes that all ’s commute with and hence demands that (if the proportionality coefficients are complex we absorb the phases into a redefinition of the noise). Thus, we set where the coefficient , with the dimension of a frequency (inverse of time), is going to be interpreted as the friction coefficient. The quantum SDE, defining the quantum stochastic Heisenberg XXZ model, is then

(18) |

with and . Again we have introduced a control dimensionless parameter . Because of the remarkable relations and , the equations of motion (18) for the density and the current are:

(19) |

with .

As pointed out in Section 2, this quantum SDE is actually a stochastic unitary evolution with

(20) |

The evolution equation for the density matrix, , can be written in a stochastic Lindblad form:

(21) |

with, again, and . Let us insist that, for each realization of the Brownian motions, the density matrix evolution is unitary, but its mean (w.r.t. to the Brownian motions) is dissipative.

This model can of course be generalized by including inhomogeneities. This amounts to replace the hamiltonian generator by , with real numbers controlling the strength of the noise independently on each site.

We could also have introduced variants of the model by changing the coupling between the noise and the spin chain degrees of freedom. Besides the previous definition another natural choice would have been to couple the noise and the spin chain via the local energy density. The hamiltonian would then have been with the local energy density. But this model is more difficult to solve because the ’s do not commute.

To simplify the following discussion we restrict ourselves to the simple homogeneous coupling. In order to ease the reading, we repeat some of the general argument presented in the previous Section – even though this may induce a few (tiny) repetitions.

### 3.2 The mean diffusive dynamics of the stochastic XXZ model

Let us start by discussing the mean dynamics and its large friction limit. The equations of motion for the mean density and the mean current are the following dissipative equations:

with and . Their structures are similar to those of the classical MFT, see eq.(2). The dissipative processes coded by the Lindbladian effectively induce current friction with a friction coefficient proportional to . The formal large friction limit, , imposes the operator constraint , which may be thought as a possible quantum analogue of Fick’s law. Of course they do not form a closed set of equations.

The mean density matrix evolves dissipatively through , or explicitly

The mean dynamics has been studied in ref.[24]. The unique steady state, which is reached at infinite time, is the uniform equilibrium state proportional to . The effective hydrodynamics, i.e. the limit at fixed, describes how this equilibrium state is attained asymptotically.

At large the mean flow is dominated by the noisy dissipative processes generated by . It converges to locally -invariant states, i.e. to states in , because the relaxation time for this dissipative process is proportional to . The Lindbladian is a sum of local terms, with . The ’s commute among themselves and are all negative operators. The kernel of each are spanned by the identity and . Thus the locally -invariant states are the density matrices with local components diagonal in the ’s basis. For instance, if we assume factorization, they are of the form . But a general locally -invariant state may not be factorized. These are the mean slow modes. Let us denote them .

As explained above in Section 2.2, since the mean slow modes form a high dimensional manifold they undergo a slow dynamical evolution (w.r.t. to the slow time ). This mean slow dynamics is determined via a second order perturbation theory. It reads with for , with and the inverse of the restriction of to the complement of and the projector on . Peculiar properties of the space of operators, of the Heisenberg hamiltonian, and especially of the Lindbladian , allow us to show that, in this particular case, the operator simplifies to :

(22) |

or equivalently, thanks to the specific form of ,

(23) |

This is clearly a dissipative, Lindblad form, evolution coding for incoherent left / right hoping along the chain (which, as a model of incoherent hoping, could have been written directly without our journey through the stochastic XXZ model). It is independent of . See the Appendix F for details.

Equation (23) is a diffusive equation (it involves second order derivatives in the form of double commutators). The slow evolution of the local spins reads . As shown in the Appendix F, it reduces to :

(24) |

This is indeed a simple discrete diffusion equation (independently of the anisotropy parameter ) with a diffusion constant inversely proportional to the friction coefficient, as expected from the classical considerations of Section 1.

### 3.3 The XXZ stochastic slow modes

Equation (23) describes the mean slow mode evolution. There are of course fluctuations, which we now describe. For any given realization of the Brownian motions, the evolution equation for the density matrix is

with with the XXZ hamiltonian. We may test this stochastic evolution by looking at the mean of any function of the density matrix. For instance, we may consider polynomial functions, say , and look at their mean. This amounts to look for statistical correlations between operator expectations. Let be their expectations (w.r.t. the the Brownian motions) of those functions. Their evolutions are coded in a Fokker-Planck like equation of the form

with a second order differential operator. It decomposes into where is the Fokker-Planck operator associated to the noisy dynamics generated by the stochastic hamiltonian and is the first order differential operator associated to the hamiltonian dynamics generated by the XXZ hamiltonian . The explicit expression of those differential operators are given in Appendix E.

Let us now identify what the slow modes are. These modes are those whose expectations are non trivial in the large friction limit at fixed slow time . It is clear that the functions which are not in the kernel of , i.e. those such that , have expectations which decrease exponentially fast in time with a time scale of order – because their evolution equations are of the form where stand for sub-leading terms in . Functions which are annihilated by are those which are invariant under all local s generated by the ’s. That is: are made of s invariant functions. Let be the projector on s invariant functions. Perturbation theory then tells us that the induced dynamics on is where the dots refer to sub-leading terms in . Hence, s invariant functions which are not in the kernel of also have a vanishing expectation in the limit at fixed time . Recall that is the differential operator associated to the hamiltonian dynamics generated by . Since where is s invariant but is not, functions in are the s invariant functions which are also invariant under the flow generated by .

In summary, the slow mode observables are the functions of the density matrix which are invariant under all the local s generated by the ’s and which are also invariant under the global generated by . By construction, these functions are those invariant under conjugacy

(25) |

for any real parameters and ’s. These functions are those which have non vanishing expectations in the large friction limit . For instance, products of local density expectations, say , are slow mode functions. But these are not the only the ones: more globally invariant functions can be constructed using the projectors defined in the following section (see below and Appendix E).

### 3.4 The effective stochastic slow dynamics of the stochastic XXZ model

Let us now determine the effective stochastic dynamics of the slow mode observables in the large friction limit (i.e. limit at fixed ). Because the slow mode functions are made of functions invariant under conjugacy by the ’s and , we can describe their dynamics using an interaction representation. Let us define by

(26) |

Going to this interaction representation is a way to absorb all the fast modes. By construction, if is a slow mode function then . So we can describe the time evolution of by looking at that of .

The evolution equation for is obtained from that of by conjugacy. Since the later is the stochastic unitary evolution generated by with , we get

(27) |

where has been defined in eq.(26).

The aim of this Section is to describe what the hydrodynamic large friction limit is. As shown in Appendix C, it reduces to the stochastic unitary evolution,