Quantum Fluctuations in Mesoscopic Systems

Quantum Fluctuations in Mesoscopic Systems

F. Benatti, F. Carollo, R. Floreanini  H. Narnhofer Dipartimento di Fisica, Università di Trieste, Trieste, 34151 Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34151 Trieste, Italy School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK Institut für Theoretische Physik, Universität Wien, A-1091, Vienna, Austria benatti@ts.infn.it, Federico.Carollo@nottingham.ac.uk, floreanini@ts.infn.it, heide.narnhofer@univie.ac.at
Abstract

Recent experimental results point to the existence of coherent quantum phenomena in systems made of a large number of particles, despite the fact that for many-body systems the presence of decoherence is hardly negligible and emerging classicality is expected. This behaviour hinges on collective observables, named quantum fluctuations, that retain a quantum character even in the thermodynamic limit: they provide useful tools for studying properties of many-body systems at the mesoscopic level, in between the quantum microscopic scale and the classical macroscopic one. We hereby present the general theory of quantum fluctuations in mesoscopic systems and study their dynamics in a quantum open system setting, taking into account the unavoidable effects of dissipation and noise induced by the external environment. As in the case of microscopic systems, decoherence is not always the only dominating effect at the mesoscopic scale: certain type of environments can provide means for entangling collective fluctuations through a purely noisy mechanism.

Keywords: quantum fluctuations, mesoscopic systems, open quantum dynamics, entanglement

1 Introduction

When dealing with quantum systems formed by a large number of elementary constituents, the study of their microscopic properties becomes impractical, due to the high multiplicity of the basic elements. Instead, collective observables, i.e. observables involving all system degrees of freedom, can be directly connected to measurable quantities, and therefore constitute the most suited operators to be used to describe the physical properties of such many-body systems. Collective observables are of extensive character, growing indefinitely as the number of microscopic constituents becomes large: they need to be normalized by suitable powers of in order to obtain physically sensible definitions. In this way, provided the system density is kept fixed, these normalized collective observables become independent from the number of constituents, allowing oneself to work in the so-called thermodynamic, large limit [1]-[4].

Typical examples of collective observables are provided by the so-called mean-field operators: they are averages over all constituents of single particle quantities, an example of which is the mean magnetization in spin systems. Although the single particle observables possess a quantum character, mean-field observables show in general a classical-like behaviour as the number of constituents increases, thus becoming examples of the so-called macroscopic observables. The well-established mean-field approach to the study of many-body systems precisely accounts for their behaviour at this macroscopic, semiclassical level, where very little, if none, quantum character survives.

It thus came as a surprise the report of having observed coherent quantum behaviour also in systems made of a large number of particles [5]-[18], typically, involving Bose-Einstein condensates, namely thousands of ultracold atoms trapped in optical lattices [19]-[28], hybrid atom-photon [29]-[39] or optomechanical systems [40]-[52], where decoherence effects can hardly be neglected and emerging classicality is ultimately expected. Mean-field observables can not be used to explain such a behaviour: as mentioned, being averages quantities, scaling as for large , they show a semiclassical character. However, other kinds of collective observables have been introduced and studied in many-body systems [53]-[56]; they account for the variation of microscopic quantities around their averages computed with respect to a chosen reference state: in analogy with classical probability theory, they are called quantum fluctuations. These observables still involve all system degrees of freedom; however, scaling as with the number of constituents, they retain some quantum properties even in the thermodynamic limit. Being half-way between the microscopic observables, those describing the behaviour of single particles in the system, and the macroscopic mean-field observables, they are named mesoscopic. Indeed, quantum fluctuations always form noncommutative algebras, thus providing a useful tool for analyzing those quantum many-body properties that persist at an intermediate scale, in between the microscopic world and the classical macroscopic one.

One of the most striking manifestation of a quantum behaviour is the possibility of establishing correlations between parts of a physical system that have no classical analog, i.e. of generating entanglement between them [57]-[61]. At first considered as a mere curiosity, quantum correlations and entanglement have nowadays become physical resources allowing the realization of protocols and tasks in quantum information technologies not permitted by purely classical means [62, 63].

Entanglement is however an extremely fragile resource, that can be rapidly depleted by the action of an external environment. In general, any quantum system, and in particular a many-body one, can hardly be considered to be completely isolated: coupling to its surroundings is unavoidable and, generically, this leads to noisy and decoherence effects, eventually washing away any quantum behaviour [64]-[81].

Nevertheless, it has been found that an external environment can be responsible not only for degrading quantum coherence and entanglement, but, quite surprisingly, also of enhancing quantum correlations through a purely mixing mechanism. Indeed, it has been shown that, in certain circumstances, two independent, non interacting systems can become entangled by the action of a common bath in which they are immersed. In general, the obvious way of entangling two quantum systems is through a direct interaction among them; a different possibility is to put them in contact with an external environment: the presence of the bath induces a mixing-enhancing mechanism able to actually generate quantum correlations among them [82]-[93]. This interesting effect has been proven to occur in microscopic systems, made of two qubits or oscillators; surprisingly, it works at the mesoscopic scale also in many-body systems, provided one focuses on suitably chosen fluctuation observables.

Aim of this report is to give an overview of the theory of quantum fluctuations in reference to quantum correlations and entanglement in open many-body quantum systems at the mesoscopic scale.

Observables having the form of fluctuations were first introduced in the late 1980’s in the analysis of quantum lattice systems with short-range interactions [53]-[55]. There, it was observed that the set of all these fluctuation observables form an algebra, that, irrespective of the nature of the microscopic constituents, turns out to be nonclassical, i.e. noncommutative, and always of bosonic character: it is at the elements of this algebra that one should look in order to properly describe quantum features of many-body systems at the mesoscopic scale. These results proved to be very useful in understanding the basis of the linear response theory and the Onsager relations [94]-[96], and started extensive studies on the characteristics and basic time evolution of the fluctuation operator algebra in various physical models [97]-[117].

Despite these successes, since recently very little was known of the behaviour of quantum fluctuations in open many-body systems, i.e. in systems in contact with an external environment: this is the most common situation encountered in actual experiments, that can never be thought of as completely isolated from their surroundings. Taking as reference systems models made of a collection of either spins or oscillators immersed in a common bath, a comprehensive analysis of open, dissipative dynamics of many-body fluctuation operators can be given [118]-[125]. With respect to the unitary time evolutions explored so far, the presence of the external environment poses specific challenges in the derivation of the mesoscopic dynamics, leading however to interesting new physical results: two non-interacting many-body systems in a common bath can become entangled at the level of mesoscopic fluctuations, and, in certain situations, the created quantum correlations can persisit even for asymptotic long times.

Of particular interest is the application of the theory of quantum fluctuations to models with long-range interactions [126]-[133], shedding new light on the physical properties of such many-body systems at the mesoscopic scale. In these cases, the microscopic dynamics is implemented through mean-field operators, i.e. with interaction and dissipative terms scaling as ; in the thermodynamic limit, it converges to a non-Markovian [134]-[137], unitary dynamics on local operators, while giving rise to a non-linear, dissipative dynamics at the level of quantum fluctuations.


In detail the structure of the review is as follows.

In the following Section, the basic mathematical tools for the description of many-body quantum systems are briefly reviewed: they are based on the algebraic approach to quantum mechanics, which represents the most general formulation of the theory, valid for both finite and infinite dimensional systems [138]-[149]. The characteristic properties of collective many-body observables, and in particular quantum fluctuations, are subsequently discussed: in presence of short-range correlations, in the thermodynamic limit, fluctuation operators are seen to become bosonic quantum variables with Gaussian characteristic function [150]-[160]. Such a limiting behaviour is rooted in the extension to the quantum setting of the classical central limit theorem [161, 162]. These abstract results are then applied to the discussion of many-body systems composed by spin-chains or collections of independent oscillators.

Section 3 is instead devoted to the study of the dynamics of quantum fluctuations. The focus is on open, dissipative time evolutions as given by microscopic, local generators in Kossakowski-Lindblad form [73]-[77]. Under rather general conditions, one can show that the emergent, large mesoscopic dynamics for the bosonic fluctuations turns out to be a quantum dynamical semigroup of quasi-free type, thus preserving the Gaussian character of the fluctuation algebra. When dealing with bipartite many-body systems, this emergent dissipative Gaussian dynamics is able to create mesoscopic entanglement at the level of fluctuation operators through a purely noisy mechanism, namely, without environment mediated interaction among the mesoscopic degrees of freedom. Remarkably, in certain situations, the generated entanglement can persist for asymptotic long times. The behaviour of the created collective quantum correlations can be studied as a function of the characteristics of the external environment in which the mesoscopic system is immersed. One then discovers that a sort of entanglement phase transition is at work: a critical temperature can always be identified, above which quantum correlations between mesoscopic observables can not be created.

Section 4 deals with systems with long-range interactions [130]-[133]. In the thermodynamic limit, the dissipative dynamics of such systems behaves quite differently depending on whether one focuses on microscopic or collective observables. Quite surprisingly, the time evolution of local, i.e. microscopic, observables turns out to be an automorphism of non-Markovian character, generated by a time-dependent Hamiltonian, while that of quantum fluctuations, i.e. of mesoscopic observables, consists of a one-parameter family of non-linear maps. These maps can be extended to a larger algebra in such a way that their generator becomes time-independent, giving rise to a semigroup of completely positive maps, whose generator is however of hybrid type, containing quantum as well as classical contributions.

Finally, let us point out that the theory of quantum fluctuations is very general and independent from the specific models here discussed. In this respect, it can be applied in all instances where mesoscopic, coherent quantum behaviours are expected to emerge, e.g. in experiments involving spin-like and optomechanical systems, or trapped ultra-cold atom gases: the possibility of entangling these many-body systems through a purely mixing mechanism may reinforce their use in the actual realization of quantum information and communication protocols.


2 Many-body collective observables

We shall consider quantum systems composed by (distinguishable) particles and analyze their behavior in the the so-called thermodynamic, large limit by studying their collective properties.

The proper treatment of infinite quantum systems requires the use of the algebraic approach to quantum physics: in the coming subsection, we shall briefly summarize its main features, underlying the concepts and tools that will be needed in the following discussions. [For a more detailed presentation, see the reference textbooks [138]-[142].]

2.1 Observables and states

Any quantum system can be characterized by the collections of observations that can be made on it through suitable measurement processes [143]. The physical quantities that are thus accessed are the observables of the system, forming an algebra under multiplication and linear combinations, the algebra of observables.


-algebras In general, the algebra turns out to be a non-commutative -algebra; this means that it is a linear, associative algebra (with unity) over the field of complex numbers , i.e. a vector space over , with an associative product, linear in both factors. Further, is endowed with an operation of conjugation: it posses an antilinear involution , such that , for any element of . In addition, a norm is defined on , satisfying , for any (thus implying that the product operation is continuous), and such that , so that ; moreover, is closed under this norm, meaning that is a complete space with respect to the topology induced by the norm (a property that in turn makes a Banach algebra).

In the case of an -level system, can be identified with the -algebra of complex matrices; the -operation coincides now with the hermitian conjugation, , for any element , while the norm is given by the square root of the largest eigenvalue of . Nevertheless, the description of a physical system through its -algebra of observables is particularly appropriate in presence of an infinite number of degrees of freedom, where the canonical formalism is in general problematic.


States on -algebras Although the system observables, i.e. the hermitian elements of , can be identified with the physical quantities measured in experiments, the explicit link between the algebra and the outcome of the measurements is given by the concept of a state , through which the expectation value of the observable can be defined.

In general, a state on a -algebra is a linear map , with the property of being positive, i.e. , , and normalized, , indicating with the unit of . It immediately follows that the map is also continuous: , for all .

This general definition of state of a quantum system comprises the standard one in terms of normalized density matrices on a Hilbert space ; indeed, any density matrix defines a state on the algebra of bounded operators on through the relation

(1)

which for pure states, , reduces to the standard expectation: . Nevertheless, the definition in terms of is more general, holding even for systems with infinitely many degrees of freedom, for which the usual approach in terms of state vectors may be unavailable.

As for density matrices on a Hilbert space , a state on a -algebra is said to be pure if it can not be decomposed as a convex sum of two states, i.e. if the decomposition , with , holds only for . If a state is not pure, it is called mixed. It is worth noticing that, for consistency, the assumed completeness of the relation between observables and measurements on a physical system requires that the observables separate the states, i.e. for all implies , and similarly that the states separate the observables, i.e. for all states on implies .


GNS-Construction Although the above description of a quantum system through its -algebra of observables (its measurable properties) and states over it (giving the observable expectations) looks rather abstract, it actually allows an Hilbert space interpretation, through the so-called Gelfang-Naimark-Segal(GNS)-construction.

GNS Theorem Any state on the -algebra uniquely determines (up to isometries) a representation of the elements of as operators in a Hilbert space , containing a reference vector , whose matrix elements reproduce the observable expectations:

(2)

This result makes apparent that the notion of Hilbert space associated to a quantum system is not a primary concept, but an emergent tool, a consequence of the -algebra structure of the system observables. We shall now apply these basic algebraic tools to the description of many-body quantum systems.

2.2 Quasi-local algebra

Being distinguishable, each particle in the many-body system can be identified by an integer index . In view of the previous discussion, its physical properties can be described by the algebra of single-particle observables, that will be assumed to be the same algebra for all particles. When its dimension is finite, can be identified with ; nevertheless, it can be also infinite-dimensional (e.g. the oscillator algebra).

Referring to different degrees of freedom, operator algebras of different particles commute: , . By means of the tensor product structure one can construct local algebras, referring just to a finite number of particles. For instance, the algebra

(3)

contains all observables pertaining to the set of particles whose label is between and . The family of local algebras possesses the following properties [138]:

One then consider the union of these algebras over all possible finite sets of particles, , and its completion with respect to the norm inherited from the local algebras. The resulting algebra is called the quasi-local algebra: it contains all the observables of the system. In the following, generic elements of will be denoted with capital letters, , while lower case letters, , will represent elements of . Actually, any observable of particle can be embedded into as

(4)

where in the above infinite tensor product of identity operators, appears exactly at position . As a result, acts non trivially only on the -th particle. Furthermore, some operators in the quasi-local algebra act non-trivially only on a finite set of particles: they will be called (strictly) local operators. Since is the norm closure of the union of all possible local algebras, the set of all its local elements is dense; in other terms, any element of can be approximated (in norm) by local operators, with an error that can be made arbitrarily small.

States for the system will be described by positive, normalized, linear functionals on : they assign the expectation value to any operator . In the following, we shall restrict the attention to states for which the expectation values of a same observable for different particles coincide:

(5)

In other terms, the mean value of single-particle operators are the same for all particles; to remark this fact, we shall use the simpler notation:

(6)

When the single-particle algebra is finite dimensional, recalling (1), one can further write: , with a single-particle density matrix.

In addition to property (5), called translation invariance, we shall require that the states of the system to be also clustering, i.e. not supporting correlations between far away localized operators:

(7)

where is the spacial translation operator.

Using this algebraic setting, we shall see that the common wisdom that assigns a “classical” behaviour to operator averages while a non-trivial dynamics to fluctuations holds also in the case of quantum many-body systems. More specifically, mean-field observables will be shown to provide a classical (commutative) description of the system, typical of the “macroscopic” world, while fluctuations around operator averages will still retain some quantum (noncommutative) properties: they describe the “mesoscopic” behaviour of the system, at a level that is half way between the microscopic and macroscopic scale.

2.3 Mean-field observables

Single-particle operators, or more in general local operators, are observables suitable for a microscopic description of a many-body system. However, due to experimental limitations, these operators are hardly accessible in practice; only, collective observables, involving all system particles, are in general available to the experimental investigation.

In order to move from a microscopic description to a one involving collective operators, potentially defined over system with an infinitely large number of constituents, a suitable scaling needs to be chosen. The simplest example of collective observables are mean-field operators, i.e. averages of copies of a same single site observable :

(8)

We are interested in studying their behaviour in the thermodynamic, large limit.

As a first, preliminary step, let us consider two such operators, and , constructed from single-particle observables and , respectively, and compute their commutator:

(9)

where the last equality comes from the fact that operators referring to different particles commute. Since is clearly itself a a mean-field operator, one realizes that the commutator of two mean-field operators is still a mean-field operator, although with an additional factor; because of this extra factor, it vanishes in the large limit. In other terms, mean-field operators seem to provide only a “classical”, commutative description of the many-body system, any quantum, non-commutative character being lost in the thermodynamic limit.

The above result actually holds in the so-called weak operator topology [138], i.e. under state average. More precisely, for a clustering state , one has:

(10)

Indeed, for any integer one can write:

Clearly, the first piece in the r.h.s. gives no contributions in the limit. Concerning the second term, we can appeal to the fact that local operators are norm dense in ; then, without loss of generality, one can assume to be large so that involves only particles with labels . Recalling the clustering property (7), one then immediately gets the result (10). This means that, in the weak operator topology, the large limit of is a scalar multiple of the identity operator:

With similar manipulations, one can also prove that the product of two mean-field-observables weakly converges to [120]:

(11)

Furthermore, under the stronger -clustering condition (see next Section and [56]),

(12)

the following scaling can be proven [120]:

(13)

It thus follows that the weak-limit of mean-field observables gives rise to a commutative (von Neumann) algebra.

Therefore, mean-field observables describe what we can call “macroscopic”, classical degrees of freedom; although constructed in terms of microscopic operators, in the large limit they do not retain any fingerprint of a quantum behaviour. Instead, as remarked in the Introduction, we are interested in studying collective observables, involving all system particles, showing a quantum character even in the thermodynamic limit. Clearly, a less rapid scaling than is needed.

2.4 Quantum fluctuations

Fluctuation operators are collective observables that scale as the square root of and represent a deviation from the average. Given any single-particle operator and a reference state , its corresponding fluctuation operator is defined as

(14)

it is the quantum analog of a fluctuation random variable in classical probability theory [163].

Although the scaling does not in general guarantee convergence in the weak operator topology, one can make sense of the large limit of (14) in some state-induced topology. Indeed, note that the mean value of the fluctuation always vanishes: . Moreover, one has:

(15)

so that for states satisfying the -clustering condition, introduced earlier in (12), the variance of the fluctuations is bounded in the limit of large .

In addition, fluctuation operators retain a quantum behaviour in the large limit. Consider two single-particle operators and call their commutator. Since , following steps similar to the one used in the proof of (10), one can write for a clustering state :

with and arbitrary elements of . Thus, commutators of fluctuations of local operators give rise to mean-field observables, and as such, behave for large as scalar multiples of the identity, . In other terms, in the thermodynamic limit fluctuations provide commutation relations that look like standard canonical bosonic ones. These results indicate that, at the mesoscopic level, a non-commutative bosonic algebraic structure naturally emerges: quantum fluctuations indeed form a so-called quantum fluctuation algebra.

In order to explicitly construct this algebra, one starts by considering the set of self-adjoint elements of the quasi-local algebra . Actually, as shown by the examples presented below, only subsets of this set are in general physically relevant, so that one can limit the discussion to one of them. Let us then fix a set of linearly independent, self-adjoint elements in the single-particle algebra and consider their real linear span:

(16)

Following the definition (14), one can then construct the fluctuation operators corresponding to , , and the one corresponding to the generic combination , obtained from those by linearity:

(17)

we want to study the large behaviour of these fluctuation operators, having fixed a state satisfying the invariance and clustering properties in (5) and (7).

In order to build well behaved fluctuations, the discussion leading to (15) suggests to choose observables for which the -clustering property (12) is satisfied for all elements of the space . This condition guaranties that the correlation matrix , with components:

(18)

be well defined [56]. This matrix can be decomposed as

(19)

in terms of the covariance matrix, namely its real, symmetric part , with components

(20)

with indicating anticommutator, and its imaginary, antisymmetric part , with components:

(21)

Although this matrix need not be invertible, it is usually called the symplectic matrix [56]. Indeed, for a non-degenerate , the real -dimensional space becomes a symplectic space.111When is not invertible, one can restrict the discussion to a suitable, physically relevant subspace of for which the restricted becomes non-degenerate (e.g. see Sect.2.5.1 below). As such, it supports a bosonic algebra , defined as the complex vector space generated by the linear span of operators , with , obeying the following algebraic relations:

(22)
(23)

These relations are just a generalization of the familiar commutation relations of Weyl operators constructed with single-particle position and momentum operators; for this reason the unitary operators are called (generalized) Weyl operators, and the algebra they generate, a (generalized) Weyl algebra [138, 147]. As for any operator algebra, a state on this Weyl algebra is a positive, normalized linear functional , assigning its mean value to any element of the algebra. The so-called quasi-free states form an important class of such states: they are characterized by giving a mean value to Weyl operators in Gaussian form [152, 153],

(24)

The covariance is a positive, symmetric matrix, which, together with the symplectic matrix, obeys the condition

(25)

thus assuring the positivity of . Quasi-free states are regular states,222 A state on the Weyl algebra is called regular if for any real constant the map is continuous, for all [138]. Also irregular states of Weyl algebras have interesting physical applications; for a recent account, see [144]. and as such they admit a representation in terms of Bose fields. Let us denote by the GNS-representation based on the quasi-free state ; then, in this representation, the Weyl operators can be expressed as:

(26)

in terms of (unbounded) Bose operators , . They provide an explicit expression for the associated covariance matrix as their anticommutator:

(27)

while, thanks to the algebraic relation (22), their commutator gives the symplectic matrix:

(28)

The analogy of the relations (27) and (28) with the results (20) and (21) suggests to consider elements in the quasi-local algebra obtained by exponentiating the fluctuations in (17),

(29)

and focus on states for which the expectation becomes Gaussian in the large limit. The operators will be called Weyl-like operators as they behave as true Weyl operators in the thermodynamic limit. Indeed, let us consider the product of two Weyl-like operators; using the Baker-Campbell-Hausdorff formula, we can write:

As already seen, in the large limit, the first commutator on the r.h.s. is proportional to the identity, while all the additional terms vanish in norm; for instance, one has

Therefore, in the thermodynamic limit the Weyl-like operators are seen to obey the following algebraic relations:

(30)

which, recalling (21), reduce to the Weyl relations (22). In other terms, under suitable conditions, in the large limit the operators behave as the Weyl operators of the algebra . The precise way in which this statement should be understood is provided by the following result:

Theorem 1

Given the quasi-local algebra and the real linear vector space as in (16), and a clustering state on , satisfying the conditions:

(31)
(32)

one can define a Gaussian state on the Weyl algebra such that, for all , ,

(33)

with

(34)

Notice that the Gaussian state on the algebra , with covariance matrix , is indeed a well defined state. First of all, it is normalized as easily seen by setting in (34). Further, its positivity is guaranteed by the positivity of the correlation matrix (18):

Being Gaussian, the state gives rise to a regular representation of the Weyl algebra , so that one can introduce the Bose fields as in (26) and, through (29) and (34), i.e. , identify the large limit of local fluctuation operators with those Bose fields:

(35)

Let us stress that these fields, despite being collective operators, retain a quantum, non-commutative character. They describe the behaviour of many-body systems at a level that is half way between the microscopic world of single-particle observables and the macroscopic realm of mean-field operators discussed earlier. In this respect, the large limit that allows to pass from the exponential (29) of the local fluctuations (14) to the mesoscopic operators belonging to the Weyl algebra , as described by the previous Theorem, can be called the mesoscopic limit. It can be given a formal definition:


Mesoscopic limit. Given an operator , linear combination of exponential operators , we shall say that it possesses the mesoscopic limit , writing

if and only if

(36)

Note that, by varying , the expectation values of the form completely determine any generic operator in the Weyl algebra : essentially, they represent its corresponding matrix elements.333In more precise mathematical terms, the r.h.s of (36) corresponds to the matrix elements of the operator with respect to the two vectors , in the GNS-representation of the Weyl algebra based on the state [138]. Since these vectors are dense in the corresponding Hilbert space, those matrix elements completely define the operators .

Similar considerations can be formulated concerning the dynamics of many-body systems at the mesoscopic level. More precisely, given a one-parameter family of microscopic dynamical maps on the quasi-local algebra , we will study its action on the Weyl-like operators , in the limit of large . In other terms, we shall look for the limiting mesoscopic dynamics acting on the elements of the Weyl algebra . In line with the previously introduced mesoscopic limit, to which it reduces for , we can state the following definition:


Mesoscopic dynamics. Given a family of one-parameter maps , we shall say that it gives the mesoscopic limit on the Weyl algebra ,

if and only if

(37)

for all .


2.5 Spin and oscillator many-body systems

In order to make more transparent the definitions and results so far presented, we shall now briefly consider physically relevant models in which the whole treatment can be made very explicit.

2.5.1 Spin chain.

A paradigmatic example of a many-body system, often discussed in the literature, is given by a chain of 1/2 spins. The microscopic description of the system involves three operators , and , obeying the -algebra commutation relations:

(38)

Together with the identity operator , they generate the single-spin algebra , which in this particular case can be identified with , the set of all complex matrices; this algebra is attached to each site of the chain. The tensor product of single-site algebras from site to site , , as in (3), forms the local algebras . The union of these local algebras over all possible finite sets of sites, together with its completion, gives the quasi-local algebra : it contains all the observables of the spin chain.

We shall equip with a thermal state , at temperature , constructed from the tensor product of single-site thermal states:

(39)

At the generic site , the state is determined by its expectation on the basis operators:

(40)

It can be represented by a Gibbs density matrix constructed with the site- Hamiltonian

(41)

so that for any operator :

(42)

For a chain containing a finite number of sites, the state in (39) can similarly be represented by a density matrix as:

(43)

However, this is not longer possible in the thermodynamic limit; indeed, although is always normalized for any , it becomes ill-defined in the large limit, since it converges (in norm) to zero:

In other terms, states of infinitely long chains can not in general be represented by density matrices; on the other hand, the definition in (39) is perfectly valid in all situations.

Given the single-site spin operators and the state , one can now construct the corresponding fluctuations as in (14):

(44)

From them, the symplectic matrix in (21) can be easily computed; taking into account the tensor product structure of the state , it reduces to the expectation of the commutator of single-site operators:

(45)

so that, explicitly:

(46)

Recalling (28), this matrix reproduces the commutators of the Bose operators obtained as mesoscopic limit of the three fluctuations (44); as a result, commutes with all remaining operators and therefore it represents a classical, collective degree of freedom. On the contrary, the two suitably rescaled operators and obey standard canonical commutations: , from which standard Weyl operators can be defined. The corresponding Weyl algebra is equipped with a quasi-free state ,

(47)

which is again a thermal state: it can be represented by a standard Gibbs density matrix:

(48)

in terms of the free Hamiltonian

(49)

2.5.2 Harmonic chain.

As a second example of many-body system, let us consider a chain of independent, free harmonic oscillators: the oscillator attached to site is described by the position and momentum variables; these operators obey standard canonical commutation relations, , so that the single-site algebra is now the Heisenberg algebra. The union of all these algebras for all sites gives the corresponding quasi-local algebra , that is usually called the oscillator algebra: elements of this algebra are polynomials in all variables , .

As in the previous example, we shall equip with a thermal state , of the form (39), with the single-site components represented by a Gibbs density matrix as in (42), where now:

(50)

with the oscillator frequency, taken for simplicity to be the same for all sites. The state clearly satisfies both the translation invariance condition (5) and the clustering property (7): in fact, it is a Gaussian state. In order to show this, one constructs the Weyl operators

(51)

with the vector with components , and a vector of real coefficients. Although any element of the oscillator algebra can be obtained by taking derivatives of with respect to the components of , it is preferable to deal with Weyl operators, since these are bounded operators, unlike coordinate and momentum operators. Indeed, the oscillator algebra should be really identified with the strong-operator closure of the Weyl algebra with respect to the so-called GNS-representation based on the chosen state (for details, see [146, 3, 138]). In this way, the algebra contains only bounded operators; in the following, when referring to the oscillator algebra, we will always mean the algebra constructed in this way.

The expectation of the Weyl operator is indeed in Gaussian form,

(52)

with a covariance matrix , whose components are defined through the anticommutator of the different components of :

(53)

with as in (40). Since the covariance matrix is proportional to the unit matrix, the state exhibits no correlations among different oscillators; the state is therefore completely separable, as shown by its product form in (39).

As it will be useful in the following, we shall now focus on the following two quadratic elements of the single-site algebra :

(54)

given the real, linear span , let us consider the corresponding fluctuation operators, defined as in (17):

(55)

One easily checks that the large behaviors of the average of the Weyl-like operator obtained by exponentiating these fluctuations, , is Gaussian:

(56)

where with we indicate the unit matrix in -dimension. In addition, the product of two Weyl-like operators behave as a single one:

(57)

with a symplectic matrix proportional to the second Pauli matrix . This allows defining collective position and momentum operators,