Research Article A Quantum Mermin-Wagner Theorem For A .

Advances in Mathematical Physics3 Here Δ(𝑥)𝑗 means the Laplacian in variable 𝑥 x (𝑗). Next, x stands for the cardinality of the particle configuration x (i.e., x 𝑘 when x 𝑀(𝑘) ), and the parameter 𝜅 is introducedin (17). (Symbol will be used for denoting the cardinalityof a general (finite) set; for example, Λ means the number (𝑗,𝑥) (𝑗 ,𝑦)denotes the particleof vertices in Λ.) Further, xΛconfiguration with the point 𝑥 x (𝑗) removed and point 𝑦added to x (𝑗 ).As in [1], we also consider a Hamiltonian HΛ x in anΓ\Λexternal field generated by a configuration x Γ\Λ {x (𝑗 ),𝑗 Γ \ Λ} 𝑀 Γ\Λ where Γ Γ is a (finite or infinite)collection of vertices. More precisely, we only consider x Γ\Λwith x (𝑗 ) 𝜅 (see (17) below) and set(HΛ x 𝜙) (xΛ )Γ\Λ1 [ Δ(𝑥) 𝑈(1) (𝑥)2 𝑗 Λ 𝑥 x (𝑗) 𝑗𝑗 Λ 𝑥 x (𝑗)[ 1 1 (𝑥 ̸ 𝑥 ) 𝑈(2) (𝑥, 𝑥 )2 𝑗 Λ 𝑥,𝑥 x (𝑗) 1 1 (𝑗 ̸ 𝑗 ) 𝐽 (d (𝑗, 𝑗 ))2 𝑗,𝑗 Λ 𝑉 (𝑥, 𝑥 )] 𝜙 (xΛ )𝑥 x (𝑗),𝑥 x (𝑗 )] 𝐽 (d (𝑗, 𝑗 )) 𝑗 Λ,𝑗 Γ\Λ 𝑉 (𝑥, 𝑥 ) 𝜙 (xΛ ) 𝑥 x (𝑗),𝑥 x ( )𝑗 𝜆 𝑗,𝑗 1 ( x (𝑗) 1, x (𝑗 ) 𝜅)𝑗,𝑗 Λ 𝑀𝑥 x (𝑗) (𝑗,𝑥) (𝑗 ,𝑦)V (d𝑦) [𝜙 (xΛ) 𝜙 (xΛ )] .(12)The novel elements in (11) and (12) compared with [1]are the presence of on-site potentials 𝑈(1) and 𝑈(2) and thesummand involving transition rates 𝜆 𝑗,𝑗 0 for jumps of aparticle from site 𝑗 to 𝑗 .We will suppose that 𝜆 𝑗,𝑗 vanishes if the graph distanced(𝑗, 𝑗 ) 1. We will also assume uniform boundedness:sup [𝜆 𝑗,𝑗 (𝑥, 𝑀) , 𝑗, 𝑗 Γ, 𝑥 𝑀] ;(13)in view of (1) it implies that the total exit rate 𝑗 :d(𝑗,𝑗 ) 1 𝜆 𝑗,𝑗 (𝑥, 𝑀) from site 𝑗 is uniformly bounded.These conditions are not sharp and can be liberalized.The model under consideration can be considered as ageneralization of the Hubbard model [6] (in its bosonic version). Its mathematical justification includes the following.(a) An opportunity to introduce a Fock space formalismincorporates a number of new features. For instance, afermonic version of the model (not considered here) emergesnaturally when the bosonic Fock space H(𝑖) is replaced bya fermonic one. Another opening provided by this modelis a possibility to consider random potentials 𝑈(1) , 𝑈(2) and𝑉 which would yield a sound generalization of the MottAnderson model. (b) Introducing jumps makes a step towardsa treatment of a model of a quantum (Bose-) gas whereparticles “live” in a single Fock space. For example, a systemof interacting quantum particles is originally confined to a“box” in a Euclidean space, with or without “internal” degreesof freedom. In the thermodynamical limit the box expandsto the whole Euclidean space. In a two-dimensional modelof a quantum gas one expects a phenomenon of invarianceunder space-translations; one hopes to be able to address thisissue in future publications. (c) A model with jumps can beanalysed by means of the theory of Markov processes whichprovides a developed methodology.Physically speaking, the model with jumps covers a situation where “light” quantum particles are subject to a “random” force and change their “location.” This class of modelsis interesting from the point of view of transport phenomenathat they may display. (An analogy with the famous Andersonmodel, in its multiparticle version, inevitably comes tomind; cf., e.g., [7].) Methodologically, such systems occupyan “intermediate” place between models where quantumparticles are “fixed” forever to their designated locations (asin [1]) and models where quantum particles move in the samespace (a Bose-gas, considered in [8, 9]). In particular, thiswork provides a bridge between [1, 8, 9]; reading this paperahead of [8, 9] might help an interested reader to get through[8, 9] at a much quicker pace.We would like to note an interesting problem of analysisof the small-mass limit (cf. [10]) from the point of MerminWagner phenomena.1.3. Assumptions on the Potentials. The between-sites potential 𝑉 is assumed to be of class 𝐶2 . Consequently, 𝑉 and itsfirst and second derivatives satisfy uniform bounds. Namely, 𝑥 , 𝑥 𝑀 𝑉 (𝑥 , 𝑥 ) , x 𝑉 (𝑥 , 𝑥 ) , x,x 𝑉 (𝑥 , 𝑥 ) 𝑉. (14)Here 𝑥 and 𝑥 run through the pairs of variables 𝑥, 𝑥 . A similar property is assumed for the on-site potential 𝑈(1) (here weneed only a 𝐶1 smoothness):(1) 𝑈(1) (𝑥) , x 𝑈(1) (𝑥) 𝑈 ,𝑥 𝑀.(15)Note that for 𝑉 and 𝑈(1) the bounds are imposed on their negative parts only.As to 𝑈(2) , we suppose that (a) 𝑈(2) (𝑥, 𝑥 ) when 𝑥 𝑥 𝜌,(16)

4Advances in Mathematical Physics̃(2) (𝑥, 𝑥 ) R such thatand (b) a 𝐶1 -function (𝑥, 𝑥 ) 𝑈(2) (2) ̃𝑈 (𝑥, 𝑥 ) 𝑈 (𝑥, 𝑥 ) whenever 𝜌(𝑥, 𝑥 ) 𝜌. Here 𝜌(𝑥, 𝑥 )stands for the (flat) Riemannian distance between points𝑥, 𝑥 𝑀. As a result of (16), there exists a “hard core” ofdiameter 𝜌, and a given atom cannot “hold” more than𝜅 ⌈V (𝑀)⌉V (𝐵 (𝜌))(17)The standard canonical variable associated with NΛ is activity𝑧 (0, ).The Hamiltonians (11) and (12) are self-adjoint (on thenatural domains) in H(Λ). Moreover, they are positivedefinite and have a discrete spectrum, cf. [14]. Furthermore, 𝑧, 𝛽 0, HΛ and HΛ x give rise to positive-definite traceΓ\Λclass operators GΛ G𝑧,𝛽,Λ and GΛ x G𝑧,𝛽,Λ x :Γ\ΛGΛ 𝑧NΛ exp [ 𝛽HΛ ] ,particles where V(𝐵(𝜌)) is the volume of a 𝑑-dimensional ballof diameter 𝜌. We will also use the bound̃(2) (𝑥, 𝑥 ) 𝑈(2) ,̃(2) (𝑥, 𝑥 ) , x 𝑈 𝑈 𝑥, 𝑥 𝑀.GΛ x 𝑧NΛ exp [ 𝛽HΛ x ] .Γ\Λ(18)Formally, (16) means that the operators in (11) and (12) areconsidered for functions 𝜙(xΛ ) vanishing when in the particleconfiguration xΛ {x (𝑗), 𝑗 Λ}, the cardinality x (𝑗) 𝜅for some 𝑗 Λ.The function 𝐽 : 𝑟 (0, ) 𝐽(𝑟) 0 is assumedmonotonically nonincreasing with 𝑟 and obeying the relation𝐽(𝑙) 0 as 𝑙 , where𝐽 (𝑙) sup [ 𝐽 (d (𝑗 , 𝑗 )) 1 (d (𝑗 , 𝑗 ) 𝑙) : 𝑗 Γ] [𝑗 Γ] .(19)2(20)Next, we assume that the functions 𝑈(1) , 𝑈(2) , and 𝑉 are ginvariant: 𝑥, 𝑥 𝑀 and g G,Definition 1. We will call GΛ and GΛ x the Gibbs operatorsΓ\Λin volume Λ, for given values of 𝑧 and 𝛽 (and—in the case ofGΛ x —with the boundary condition x Γ\Λ ).Γ\ΛThe Gibbs operators in turn give rise to the Gibbs states𝜑Λ 𝜑𝛽,𝑧,Λ and 𝜑Λ xΓ\Λ 𝜑𝛽,𝑧,Λ xΓ\Λ , at temperature 𝛽 1 andactivity 𝑧 in volume Λ. These are linear positive normalizedfunctionals on the 𝐶 -algebra BΛ of bounded operators inspace HΛ :𝜑Λ xΓ\Λ (A) trHΛ (RΛ xΓ\Λ A) ,RΛ (21)(2) 𝜅𝐽 (1) 𝑉) . (22)1.4. The Gibbs State in a Finite Volume. Define the particlenumber operator NΛ , with the actionNΛ 𝜙 (xΛ ) xΛ 𝜙 (xΛ ) ,xΛ 𝑀 Λ .(23)Here, for a given xΛ {x (𝑗), 𝑗 Λ}, xΛ stands for the totalnumber of particles in configuration xΛ : xΛ x (𝑗) .𝑗 Λ(24)GΛ xΓ\ΛΞ (Λ x Γ\Λ ),with Ξ (Λ x Γ\Λ ) Ξ𝑧,𝛽 (Λ x Γ\Λ )𝑉 (𝑥, 𝑥 ) 𝑉 (g𝑥, g𝑥 ) .In the following we will need to bound the fugacity (oractivity, cf. (25)) 𝑧 in terms of the other parameters of themodel(26)with Ξ (Λ) Ξ𝑧,𝛽 (Λ) trHΛ GΛ , (27)Γ\Λ 𝜅𝑈GΛ,Ξ (Λ)RΛ x 𝑈(2) (𝑥, 𝑥 ) 𝑈(2) (g𝑥, g𝑥 ) ,A BΛ ,where𝑈(1) (𝑥) 𝑈(1) (g𝑥) ,(1)Γ\Λ𝜑Λ (A) trHΛ (RΛ A) ,𝐽 sup [ 𝐽 (d (𝑗, 𝑗 )) d(𝑗, 𝑗 ) : 𝑗 Γ] . [𝑗 Γ]where Θ 𝜅𝛽 (𝑈(25)We would like to stress that the full range of variables 𝑧, 𝛽 0is allowed here because of the hard-core condition (16): it doesnot allow more than 𝜅 Λ particles in Λ where Λ stands forthe number of vertices in Λ. However, while passing to thethermodynamic limit, we will need to control 𝑧 and 𝛽.Additionally, let 𝐽(𝑟) be such that𝑧𝑒Θ 1,Γ\Λ(28) trHΛ (𝑧NΛ exp [ 𝛽HΛ x ]) .Γ\ΛThe hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ𝑧,𝛽 (Λ x Γ\Λ ) are finite; formally, these facts willbe verified by virtue of the Feynman-Kac representation.Definition 2. Whenever Λ0 Λ, the 𝐶 -algebra BΛ0 is identified with the 𝐶 subalgebra in BΛ formed by the operators0of the form A0 IΛ\Λ0 . Consequently, the restriction 𝜑ΛΛ of state 𝜑Λ to 𝐶 -algebra BΛ0 is given by00𝜑ΛΛ (A0 ) trHΛ0 (RΛΛ A0 ) ,A 0 BΛ 0 ,(29)where0RΛΛ trHΛ\Λ0 RΛ .(30)

Advances in Mathematical Physics50Operators RΛΛ (we again call them RDMs) are positive defi0nite and have trHΛ0 RΛΛ 1. They also satisfy the compatibility property: Λ0 Λ1 Λ,01RΛΛ trHΛ1 \Λ0 RΛΛ .0Γ\Λ0Λators RΛ x , with the same properties.Γ\Λ1.5. Limiting Gibbs States. The concept of a limiting Gibbsstate is related to notion of a quasilocal 𝐶 -algebra; see [14].For the class of systems under consideration, the constructionof the quasilocal 𝐶 -algebra BΓ is done along the same linesas in [1]: BΓ is the norm completion of the 𝐶 algebra (B0Γ ) lim ind𝑛 BΛ 𝑛 . Any family of positive-definite operators0RΛ in spaces HΛ0 of trace one, where Λ0 runs through finitesubsets of Γ, with the compatibility property10Λ1 Λ0 ,(32)determines a state of BΓ , see [12, 13].Finally, we introduce unitary operators UΛ0 (g), g G, inHΛ0 :UΛ 𝜙 (xΛ 0 ) 𝜙 (g 1 xΛ 0 ) ,(33)whereg 1 xΛ 0 {g 1 x (𝑗) , 𝑗 Λ0 } ,(34)g 1 x (𝑗) {g 1 𝑥 : 𝑥 x (𝑗)} .Theorem 3. Assuming the conditions listed above, for all 𝑧, 𝛽 0(0, ) satisfying (22) and a finite Λ0 Γ, operators RΛΛform a compact sequence in the trace-norm topology in HΛ0as Λ Γ. Furthermore, given any family of (finite or infinite)sets Γ Γ(Λ) Γ and configurations x Γ\Λ {x (𝑖), 𝑖 Γ \ Λ}0Λwith x (𝑖) 𝜅, operators RΛ xΓ\Λalso form a compact sequence001of trace one. Moreover, if Λ1 Λ0 and RΛ and RΛ are the01Λ0Λ1limits for RΛΛ and RΛΛ or for RΛ xand RΛ xalong the same Γ\Λ00UΛ0 (g) 1 RΛ UΛ0 (g) RΛ .(35)Accordingly, any limiting Gibbs state 𝜑 of B determined by a0family of limiting operators RΛ obeying (35) satisfies the corresponding invariance property: finite Λ0 Γ, any A BΛ0 ,and g G,𝜑 (A) 𝜑 (UΛ0 (g) 1 AUΛ0 (g)) .(36)Remarks. (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (i.e., absence of phase transitions).(2) Properties (35) and (36) are trivially fulfilled for the00limiting points RΛ and 𝜑 of families {RΛΛ } and {𝜑Λ }. However, they require a proof for the limit points of the familiesΛ0{RΛ x } and {𝜑Λ x Γ\Λ }.Γ\ΛThe set of limiting Gibbs states (which is nonempty dueto Theorem 3) is denoted by G0 . In Section 3 we describe aclass G G0 of states of 𝐶 -algebra B satisfying the FK-DLRequation, similar to that in [1].2. Feynman-Kac Representations forthe RDM Kernels in a Finite Volume2.1. The Representation for the Kernels of the Gibbs Operators.A starting point for the forthcoming analysis is the FeynmanKac (FK) representation for the kernels KΛ (xΛ , yΛ ) K𝛽,𝑧,Λ (xΛ , yΛ ) and FΛ (xΛ , yΛ ) F𝛽,𝑧,Λ (xΛ , yΛ ) of operatorsGΛ and RΛ .𝛽Definition 7. Given (𝑥, 𝑖), (𝑦, 𝑗) 𝑀 Γ, 𝑊(𝑥,𝑖),(𝑦,𝑗) denotesthe space of path, or trajectories, 𝜔 𝜔(𝑥,𝑖),(𝑦,𝑗) in 𝑀 Γ, oftime-length 𝛽, with the end-points (𝑥, 𝑖) and (𝑦, 𝑗). Formally,𝛽𝜔 𝑊(𝑥,𝑖),(𝑦,𝑗) is defined as follows:0in the trace-norm topology. Any limit point, RΛ , for {RΛΛ } orΛ00{RΛ x } as Λ Γ, is a positive-definite operator in H(Λ )Γ\ΛΓ\Λof trace one commuting with operators UΛ0 (g): g G,(31)In a similar fashion one defines functionals 𝜑ΛΛ x and oper-RΛ trHΛ0 \Λ1 RΛ ,0Theorem 6. Under the condition (22), any limiting point, RΛ ,0Λ0for {RΛΛ } or {RΛ x }, as Λ Γ, is a positive-definite operatorΓ\Λsubsequence Λ 𝑠 Γ, then the property (32) holds true.Consequently, the Gibbs states 𝜑Λ and 𝜑Λ x form compactΓ\Λsequences as Λ Γ.Remark 4. In fact, the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (Γ, E),only under an assumption that the degree of the vertices inΓ is uniformy bounded.Definition 5. Any limit point 𝜑 for states 𝜑Λ and 𝜑Λ x isΓ\Λcalled a limiting Gibbs state (for given 𝑧, 𝛽 (0, )).𝜔 : 𝜏 [0, 𝛽] 𝜔 (𝜏) (𝑢 (𝜔, 𝜏) , 𝑙 (𝜔, 𝜏)) 𝑀 Γ,́ ag;́𝜔 is cadl𝜔 (0) (𝑥, 𝑖) ,𝜔 (𝛽 ) (𝑦, 𝑗) ,𝜔 has finitely many jumps on [0, 𝛽] ;if a jump occurs at time 𝜏, then d [𝑙 (𝜔, 𝜏 ) , 𝑙 (𝜔, 𝜏)] 1.(37)The notation 𝜔(𝜏) and its alternative, (𝑢(𝜔, 𝜏), 𝑙(𝜔, 𝜏)), forthe position and the index of trajectory 𝜔 at time 𝜏 will beemployed as equal in rights. We use the term the temporalsection (or simply the section) of path 𝜔 at time 𝜏.Definition 8. Let xΛ {x (𝑖), 𝑖 Λ} 𝑀 Λ , and yΛ {y (𝑗), 𝑗 Λ} 𝑀 Λ be particle configurations over Λ, with xΛ yΛ . A matching (or pairing) 𝛾 between xΛ and yΛ is

6Advances in Mathematical Physicsdefined as a collection of pairs [(𝑥, 𝑖), (𝑦, 𝑗)]𝛾 , with 𝑖, 𝑗 Λ,𝑥 x (𝑖), and 𝑦 y (𝑗), with the properties that (i) 𝑖 Λand 𝑥 x (𝑖) : there exist unique 𝑗 Λ and 𝑦 y (𝑗)such that (𝑥, 𝑖) and (𝑦, 𝑗) form a pair, and (ii) 𝑗 Λ and𝑦 y (𝑗) : there exist unique 𝑖 Λ and 𝑥 x (𝑖) suchthat (𝑥, 𝑖) and (𝑦, 𝑗) form a pair. (Owing to the condition xΛ yΛ , these properties are equivalent.) It is convenientto write [(𝑥, 𝑖), (𝑦, 𝑗)]𝛾 [(𝑥, 𝑖), 𝛾(𝑥, 𝑖)].Next, consider the Cartesian product𝛽𝑊x ,y ,𝛾Λ Λ 𝛽𝑊(𝑥,𝑖),𝛾(𝑥,𝑖) , 𝑖 Λ 𝑥 x (𝑖)𝜏 0 starts from the point 𝑥 and has the initial index 𝑖 whileat time 𝜏 𝛽 it is at the point 𝑦 and has the index 𝑗. The value𝛽𝛽𝑝̂(𝑥,𝑖),(𝑦,𝑗) P(𝑥,𝑖),(𝑦,𝑗) (𝑊(𝑥,𝑖),(𝑦,𝑗) ) is given by𝛽𝑝̂(𝑥,𝑖),(𝑦,𝑗) 1 (𝑖 𝑗) 𝑝𝑀 (𝑥, 𝑦) exp [ 𝛽 𝜆 𝑖,𝑗̃]̃ (𝑖,𝑗̃) 1[ 𝑗:d] (38)𝛽 𝜆 𝑙𝑠 ,𝑙𝑠 1 d𝜏𝑠 exp [ (𝜏𝑠 1 𝜏𝑠 ) 𝜆 𝑙𝑠 ,𝑗̃]0̃ 𝑠 ,𝑗) 1̃𝑗:d(𝑙[]and the disjoint union𝛽𝛽𝑊x ,y 𝑊x ,y ,𝛾 .ΛΛΛ𝛾 1 (0 𝜏0 𝜏1 𝜏𝑘 𝜏𝑘 1 𝛽) ,(41)(39)Λ𝛽Accordingly, an element 𝜔Λ 𝑊x ,y ,𝛾 in (38) represents aΛ Λcollection of paths 𝜔𝑥,𝑖 , 𝑥 x (𝑖), 𝑖 Λ, of time-length 𝛽,starting at (𝑥, 𝑖) and ending up at 𝛾(𝑥, 𝑖). We say that 𝜔Λ is apath configuration in (or over) Λ.𝛽where 𝑝𝑀(𝑥, 𝑦) denotes the transition probability density forthe Brownian motion to pass from 𝑥 to 𝑦 on 𝑀 in time 𝛽:𝛽ΛΛΛΛDefinition 9. In what follows, 𝜉(𝜏), 𝜏 0, stands for theMarkov process on 𝑀 Γ, with cádlág trajectories, determined by the generator L acting on a function (𝑥, 𝑖) 𝑀 Λ 𝜓(𝑥, 𝑖) by1L𝜓 (𝑥, 𝑖) Δ𝜓 (𝑥, 𝑖)2 𝜆 𝑖,𝑗 V (d𝑦) [𝜓 (𝑦, 𝑗) 𝜓 (𝑥, 𝑖)] .𝑗:d(𝑖,𝑗) 1(2𝜋𝛽) 𝛽cylinder sets) in 𝑊(𝑥,𝑖),(𝑦,𝑗) , 𝑊x ,y ,𝛾 , and 𝑊x ,y . 1𝛽𝑝𝑀 (𝑥, 𝑦) The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consideration.We will work with standard sigma algebras (generated by𝛽𝑀(40)In the probabilistic literature, such processes are referred toas Lévy processes; see, for example, [14].Pictorially, a trajectory of process 𝜉 moves along 𝑀according to the Brownian motion with the generator Δ/2and changes the index 𝑖 Γ from time to time in accordance with jumps occurring in a Poisson process of rate 𝑗:d(𝑖,𝑗) 1 𝜆 𝑖.𝑗 . In other words, while following a Brownianmotion rule on 𝑀, having index 𝑖 Γ and being at point𝑥 𝑀, the moving particle experiences an urge to jump from𝑖 to a neighboring vertex 𝑗 and to a point 𝑦 at rate 𝜆 𝑖,𝑗 V(d𝑦).After a jump, the particle continues the Brownian motion on𝑀 from 𝑦 and keeps its new index 𝑗 until the next jump, andso on.For a given pairs (𝑥, 𝑖), (𝑦, 𝑗) 𝑀 Γ, we denote by𝛽𝛽P(𝑥,𝑖),(𝑦,𝑗) the nonnormalised measure on 𝑊(𝑥,𝑖),(𝑦,𝑗) induced𝛽 1 (d (𝑙𝑠 , 𝑙𝑠 1 ) 1) 𝑘 1 𝑙0 𝑖,𝑙1 ,.,𝑙𝑘 ,𝑙𝑘 1 𝑗 0 𝑠 𝑘by 𝜉. That is, under measure P(𝑥,𝑖),(𝑦,𝑗) the trajectory at time𝑑/2(42) 2 𝑥 𝑦 𝑛 exp ().2𝛽𝑑 𝑛 (𝑛1 ,.,𝑛𝑑 ) ZIn view of (13), the quantity 𝑝̂(𝑥,𝑖),(𝑦,𝑗) and its derivatives areuniformly bounded: 𝑝̂(𝑥,𝑖),(𝑦,𝑗) , 𝑥 𝑝̂(𝑥,𝑖),(𝑦,𝑗) , 𝑦 𝑝̂(𝑥,𝑖),(𝑦,𝑗) 𝑝̂𝑀,(43)𝑥, 𝑦 𝑀, 𝑖, 𝑗 Γ,where 𝑝̂𝑀 𝑝̂𝑀(𝛽) (0, ) is a constant.We suggest a term “non-normalised Brownian bridgewith jumps” for the measure but expect that a better term willbe proposed in future.Definition 10. Suppose that xΛ {x (𝑖), 𝑖 Λ} 𝑀 Λ andyΛ {y (𝑗), 𝑗 Λ} 𝑀 Λ are particle configurations overΛ, with xΛ yΛ . Let 𝛾 be a pairing between xΛ and yΛ .𝛽Then P x ,y ,𝛾 denotes the product measure on 𝑊x ,y ,𝛾 :ΛΛΛ𝛽Px ,y ,𝛾 Λ𝛽 𝑖 Λ 𝑥 x (𝑖)ΛΛP(𝑥,𝑖),𝛾(𝑥,𝑖) .𝛽(44)𝛽Furthermore, Px ,y stands for the sum measure on 𝑊x ,y :ΛΛΛ𝛽𝛽Px ,y Px ,y ,𝛾 .ΛΛ𝛾ΛΛ(45)Λ𝛽According to Definition 10, under the measure Px ,y ,𝛾 ng 𝜔Λ are indethe trajectories 𝜔𝑥,𝑖 pendent components. (Here the term independence is usedin the measure-theoretical sense.)

Advances in Mathematical Physics7𝛽As in [1], we will work with functionals on 𝑊x ,y ,𝛾 repreΛ Λsenting integrals along trajectories. The first such functional,hΛ (𝜔Λ ), is given bywhere, in turn, h(𝑥,𝑖),(𝑥 ,𝑖 ) (𝜔𝑥,𝑖 , (𝑥 , 𝑖 ))𝛽Λ d𝜏 [𝑈(2) (𝑢𝑖,𝑥 (𝜏) , 𝑥 ) 1 (𝑙𝑥,𝑖 (𝜏) 𝑖 )𝑥,𝑖h (𝜔Λ ) h (𝜔𝑥,𝑖 )0𝑖 Λ 𝑥 x (𝑖)1 2h (𝑖,𝑖 ) Λ Λ𝑥 x (𝑖),𝑥 x (𝑖 )(𝑥,𝑖),(𝑥 ,𝑖 ) 𝐽 (d (𝑗, 𝑖 )) (46)𝑗 Γ:𝑗 ̸ 𝑖 𝑉 (𝑢𝑖,𝑥 (𝜏) , 𝑥 ) 1 (𝑙𝑥,𝑖 (𝜏) 𝑗)] .(51) (𝜔𝑥,𝑖 , 𝜔𝑥 ,𝑖 ) .Here, introducing the notation 𝑢𝑥,𝑖 (𝜏) 𝑢(𝜔𝑥,𝑖 , 𝜏) and𝑢𝑥 ,𝑖 (𝜏) 𝑢(𝜔𝑥 ,𝑖 , 𝜏) for the positions in 𝑀 of paths 𝜔𝑥,𝑖 𝛽𝛽𝑊(𝑥,𝑖),𝛾(𝑥,𝑖) and 𝜔𝑥 ,𝑖 𝑊(𝑥 ,𝑖 ),𝛾(𝑥 ,𝑖 ) at time 𝜏, we define𝛽h𝑥,𝑖 (𝜔𝑥,𝑖 ) d𝜏𝑈(1) (𝑢𝑖,𝑥 (𝜏)) .0(47)Next, with 𝑙𝑥,𝑖 (𝜏) and 𝑙𝑥 ,𝑖 (𝜏) standing for the indices of 𝜔𝑥,𝑖and 𝜔𝑥 ,𝑖 at time 𝜏, h(𝑥,𝑖),(𝑥 ,𝑖 ) (𝜔𝑥,𝑖 , 𝜔𝑥 ,𝑖 )The functionals hΛ (𝜔Λ ) and hΛ (𝜔Λ x Γ\Λ ) are interpreted asenergies of path configurations. Compare (2.1.4) and (2.3.8)in [1].Finally, we introduce the indicator functional 𝛼Λ (𝜔Λ ):1,{{𝛼Λ (𝜔Λ ) {{{0,𝛽It can be derived from known results [11, 15–17] (for a directargument, see [18]) that the following assertion holds true.Γ\Λ(GΛ 𝜙) (xΛ ) (GΛ x 𝜙) (xΛ )1 𝐽 (d (𝑗 , 𝑗 ))2 (𝑗 ,𝑗 ) Γ ΓΓ\Λ (53) 1 (𝑙𝑥,𝑖 (𝜏) 𝑗 ̸ 𝑗 𝑙𝑥 ,𝑖 (𝜏)) ] .](48)Next, consider the functional hΛ (𝜔Λ x Γ\Λ ): for x Γ\Λ {x (𝑗), 𝑗 Γ \ Λ}. As before, we assume that x (𝑗) 𝜅.Define(49)𝛽 KΛ (xΛ , yΛ ) 𝑧 xΛ 𝑊x ,y Λ ΛPx ,y (d𝜔Λ ) 𝛼Λ (𝜔Λ )ΛΛ(54) exp [ h (𝜔Λ )] ,KΛ (xΛ , yΛ x Γ\Λ )𝛽 𝑧 xΛ 𝑊x ,y Λ ΛhΛ (𝜔Λ x Γ\Λ ) Moreover, the integral kernels KΛ (xΛ , yΛ ) and KΛ (xΛ , yΛ x Γ\Λ ) vanish if xΛ ̸ yΛ . On the other hand, when xΛ yΛ , the kernels KΛ (xΛ , yΛ ) and KΛ (xΛ , yΛ x Γ\Λ ) admit thefollowing representations:ΛHere hΛ (𝜔Λ ) is as in (46) and V (d𝑦) KΛ (xΛ , yΛ x Γ\Λ ) 𝜙 (yΛ ) .𝑀 Λ 𝑗 Λ 𝑦 y (𝑗) 𝑉 (𝑢𝑖,𝑥 (𝜏) , 𝑢𝑖 ,𝑥 (𝜏))hΛ (𝜔Λ x Γ\Λ ) hΛ (𝜔Λ ) hΛ (𝜔Λ x Γ\Λ ) . V (d𝑦) KΛ (xΛ , yΛ ) 𝜙 (yΛ ) ,𝑀 Λ 𝑗 Λ𝑦 y (𝑗) 1 (𝑙𝑥,𝑖 (𝜏) 𝑗 𝑙𝑥 ,𝑖 (𝜏)) (52)Lemma 11. For all 𝑧, 𝛽 0 and a finite Λ, the Gibbs operatorsGΛ and GΛ x act as integral operators in H(Λ): d𝜏 [ 𝑈(2) (𝑢𝑖,𝑥 (𝜏) , 𝑢𝑖 ,𝑥 (𝜏))0 [𝑗 Γ if index 𝑙𝑥,𝑖 (𝜏) Λ, 𝜏 [0, 𝛽] , 𝑖 Λ, 𝑥 x (𝑖) ,otherwise. h(𝑥,𝑖),(𝑥 ,𝑖 )(𝑖,𝑖 ) Λ (Γ\Λ) 𝑥 x (𝑖),𝑥 x (𝑖 ) (𝜔𝑥,𝑖 , (𝑥 , 𝑖 )) ,(50)Px ,y (d𝜔Λ ) 𝛼Λ (𝜔Λ )ΛΛ exp [ hΛ (𝜔Λ x Γ\Λ )] .(55)The ingredients of these representations are determined in (46)–(51).

8Advances in Mathematical PhysicsRemark 12. As before, we stress that, owing to (16) and (17),a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 𝜔Λ {𝜔𝑥,𝑖 } such that 𝜏 [0, 𝛽] and 𝑗 Γ, the number of paths 𝜔𝑥,𝑖 with index𝑙𝑥,𝑖 (𝜏) 𝑗 is less than or equal to 𝜅. Likewise, the integral inthe RHS of (55) receives a non-zero contribution only fromconfigurations 𝜔Λ {𝜔 𝑥,𝑖 } such that, site 𝑗 Γ, the numberof paths 𝜔𝑥,𝑖 with index 𝑙𝑥,𝑖 (𝜏) 𝑗 plus the cardinality x (𝑗)does not exceed 𝜅.Definition 14. Suppose xΛ {x (𝑖), 𝑖 Λ} 𝑀 Λ and yΛ {y (𝑗), 𝑗 Λ} 𝑀 Λ are particle configurations over Λ,with xΛ yΛ . Let 𝛾 be a matching between xΛ and yΛ . Weconsider the Cartesian product: 𝑊x ,y ,𝛾 Λ 𝑖 Λ 𝑥 x (𝑖)ΛDefinition 13. For given (𝑥, 𝑖), (𝑦, 𝑗) 𝑀 Γ, the symbol 𝑊(𝑥,𝑖),(𝑦,𝑗) denotes the disjoint union: 𝑘𝛽𝑊(𝑥,𝑖),(𝑦,𝑗) 𝑊(𝑥,𝑖),(𝑦,𝑗) .(56)𝑘 0,1,. In other words, 𝑊(𝑥,𝑖),(𝑦,𝑗) is the space of paths Ω Ω(𝑥,𝑖),(𝑦,𝑗) in 𝑀 Γ, of a variable time-length 𝑘𝛽, where 𝑘 𝑘(Ω ) takesvalues 1, 2, . . . and called the length multiplicity, with the endpoints (𝑥, 𝑖) and (𝑦, 𝑗). The formal definition follows the same line as in (37), and we again use the notation Ω (𝜏) and the notation (𝑢(Ω , 𝜏), 𝑙(Ω , 𝜏)) for the pair of the position and the index of path Ω at time 𝜏. Next, we call the particle configuration {Ω (𝜏 𝛽𝑚), 0 𝑚 𝑘(Ω )} the temporal section (or simply the section) of Ω at time 𝜏 [0, 𝛽]. We also call Ω(𝑥,𝑖),(𝑦,𝑗) 𝑊(𝑥,𝑖),(𝑦,𝑗) a path (from (𝑥, 𝑖) to (𝑦, 𝑗)).A particular role will be played by closed paths (loops),with coinciding endpoints (where (𝑥, 𝑖) (𝑦, 𝑗)). Accord the set 𝑊(𝑥,𝑖),(𝑥,𝑖) . An element ofingly, we denote by 𝑊𝑥,𝑖 is denoted by Ω 𝑥,𝑖 or, in short, by Ω and called a loop𝑊𝑥,𝑖at vertex 𝑖. (The upper index indicates that the lengthmultiplicity is unrestricted.) The length multiplicity of a loop is denoted by 𝑘(Ω 𝑥,𝑖 ) or 𝑘𝑥,𝑖 . It is instructive toΩ 𝑥,𝑖 𝑊𝑥,𝑖note that, as topological object, a given loop Ω admits amultiple choice of the initial pair (𝑥, 𝑖): it can be representedby any pair (𝑢(Ω , 𝜏), 𝑙(Ω , 𝜏)) at a time 𝜏 𝑙𝛽 where 𝑙 1, . . . , 𝑘(Ω ). As above, we use the term the temporal sectionat time 𝜏 [0, 𝛽] for the particle configuration {Ω 𝑥,𝑖 (𝜏 𝛽𝑚), 0 𝑚 𝑘𝑥,𝑖 } and employ the alternative notation(𝑢(𝜏 𝛽𝑚; Ω ), 𝑙(𝜏 𝛽𝑚; Ω )) addressing the position andthe index of Ω at time 𝜏 𝛽𝑚 [0, 𝛽𝑘(Ω )].(57)and the disjoint union: 2.2. The Representation for the Partition Function. The FKrepresentations of the partition functions Ξ(Λ) Ξ𝛽,𝑧 (Λ)in (27) and Ξ(Λ x Γ\Λ ) in (1.4.6) reflect a specific characterof the traces tr GΛ and tr GΛ x in H(Λ). The source of aΓ\Λcomplication here is the jump terms in the Hamiltonians 𝐻Λand 𝐻Λ x in (11) and (12), respectively. In particular, we willΓ\Λhave to pass from trajectories of fixed time-length 𝛽 to loopsof a variable time length. To this end, a given matching 𝛾is decomposed into a product of cycles, and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-length multiple of 𝛽. (A similar constructionhas been performed in [18].)To simplify the notation, we omit, wherever possible, theindex 𝛽.𝑊(𝑥,𝑖),𝛾(𝑥,𝑖) , 𝑊x ,y 𝑊x ,y ,𝛾 .ΛΛΛ𝛾 Λ(58) Accordingly, an element ΩΛ 𝑊x ,y ,𝛾 in (58) represents aΛ Λcollection of paths Ω𝑥,𝑖 , 𝑥 x (𝑖), 𝑖 Λ, of time-length 𝑘𝛽,starting at (𝑥, 𝑖) and ending up at (𝑦, 𝑗) 𝛾(𝑥, 𝑖). We say that ΩΛ 𝑊x ,y is a path configuration in (or over) Λ.Λ ΛAgain, loops play a special role and deserve a particularnotation. Namely, 𝑊x denotes the Cartesian product:Λ𝑊x Λ 𝑖 Λ 𝑥 x (𝑖) 𝑊𝑥,𝑖,(59)and 𝑊Λ stands for the disjoint union

continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is H 1 L 2 (), where is a -dimensional unit torus R /Z with a at metric. e phase space of spins is H L sym 2 ( ), the subspace of L 2 ( ) formed by functions symm