A quantum Mermin--Wagner theorem for a generalized Hubbard model on a 2D graph

This paper is the second in a series of papers considering symmetry properties of a bosonic quantum system over an 2D graph, with continuous spins, in the spirit of the Mermin--Wagner theorem. Here we consider bosonic systems on bi-dimensional graphs where particles can jump from a vertex to another (a generalized Hubbard model). The Feynman--Kac representation is used for proving that if the local Hamiltonians are invariant under a continuous group of transformations ${\tt G}$ (a Euclidean space or a torus of dimension $d'$ acting on a torus of dimension $d\geq d'$) then any infinite-volume Gibbs state from a certain class (introduced in the paper) is also ${\tt G}$-invariant.

Abstract. This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem [14]. In the model considered here the phase space of a single spin is  We associate a space H ≃ H(i) with each vertex i ∈ Γ of a graph (Γ, E) satisfying a special bi-dimensionality property. (Physically, vertex i represents a heavy 'atom' or 'ion' that does not move but attracts a number of 'light' particles.) The kinetic energy part of the Hamiltonian includes (i) −∆/2, the minus a half of the Laplace operator on M, responsible for the motion of a particle while 'trapped' by a given atom, and (ii) an integral term describing possible 'jumps' where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U (1) (x), x ∈ M, describing a field generated by a heavy atom, (b) two-body potentials U (2) (x, y), x, y ∈ M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V (x, y), x, y ∈ M, scaled along the graph distance d(i, j) between vertices 1. Introduction 1.1. Basic facts on bi-dimensional graphs. As in [10], we suppose that the graph (Γ, E) has been given, with the set of vertices Γ and the set of edges E. The graph has the property that whenever edge (j ′ , j ′′ ) ∈ E, the reversed edge (j ′′ , j ′ ) belongs in E as well. Furthermore, graph (Γ, E) is without multiple edges and has a bounded degree, i.e. the number of edges (j, j ′ ) with a fixed initial or terminal vertex is uniformly bounded: sup max ♯ {j ′ ∈ Γ : (j, j ′ ) ∈ E}, ♯ {j ′ ∈ Γ : (j ′ , j) ∈ E} : j ∈ Γ < ∞. (1.1.1) The bi-dimensionality property is expressed in the bound 0 < sup 1 n ♯ Σ(j, n) : j ∈ Γ, n = 1, 2, . . . < ∞ (1. 1.2) where Σ(j, n) stands for the set of vertices in Γ at the graph distance n from j ∈ Γ (a sphere of radius n around j): Σ(j, n) = {j ′ ∈ Γ : d(j, j ′ ) = n}. grows at most quadratically with n.
A justification for putting a quantum system on a graph can be that graph-like structures become increasingly popular in rigorous Statistical Mechanics, e.g., in quantum gravity. Viz., see [13], [11] and [12]. On the other hand, a number of properties of Gibbs ensembles do not depend upon 'regularity' of an underlying spatial geometry.

A bosonic model in the Fock space.
With each vertex i ∈ Γ we associate a copy of a compact manifold M which we take in this paper to be a unit d-dimensional torus R d /Z d with a flat metric ρ and the volume v . We also associate with i ∈ Γ a bosonic Fock-Hilbert space H(i) ≃ H. Here H = ⊕  x * Λ ∈ M * Λ → φ(x * Λ ). Here x * Λ is a collection {x * (j), j ∈ Λ} of finite point sets x * (j) ⊂ M associated with sites j ∈ Λ. Following [10], we call x * (j) a particle configuration at site The generally accepted view is that the Hubbard model is a highly oversimplified model for strongly interacting electrons in a solid. The Hubbard model is a kind of minimum model which takes into account quantum mechanical motion of electrons in a solid, and nonlinear repulsive interaction between electrons. There is little doubt that the model is too simple to describe actual solids faithfully [28]. In our context the Hubbard Hamiltonian H Λ of the system in Λ acts as follows: (1.

2.3)
Here ∆ (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 introduced in (1.3.4). 1 Further, x * (j,x)→(j ′ ,y) Λ denotes the particle configuration with the point x ∈ x * (j) removed and point y added to x * (j ′ ).

.4) below) and set
The novel elements in (1.2.3), (1.2.4) compared with [10] are the presence of on-site potentials U (1) and U (2) and the summand involving transition rates λ j,j ′ ≥ 0 for jumps of a particle from site j to j ′ .
We will suppose that λ j,j ′ vanishes if the graph distance d(j, j ′ ) > 1. We will also assume uniform boundedness: (1.2.5) in view of (1.1.1) it implies that the total exit rate site j is uniformly bounded. These conditions are not sharp and can be liberalized.
The model under consideration can be considered as a generalization of the Hubbard model [8] (in its bosonic version). Its mathematical justification includes the following. (a) An opportunity to introduce a Fock space formalism incorporates a number of new features. For instance, a fermonic version of the model (not considered here) emerges naturally when the bosonic Fock space H(i) is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials U (1) , U (2) and V which would yield a sound generalization of the Mott-Anderson model. (b) Introducing jumps makes a step towards a treatment of a model of a quantum (Bose-) gas where particles 'live' in a single Fock space. For example, a system of interacting quantum particles are originally confined to a 'box' in a Euclidean space, with or without 'internal' degrees of freedom. In the thermodynamical limit the box expands to the whole Euclidean space. In a two-dimensional model of a quantum gas one expects a phenomenon of invariance under space-translations; one hopes to be able to address this issue in future publications. (c) A model with jumps can be analysed by means of the theory of Markov processes which provides 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 models are interesting from the point of view of transport phenomena that they may display. (An analogy with the famous Anderson model, in its multi-particle version, inevitably comes to mind; cf., e.g., [3].) Methodologically, such systems occupy an 'intermediate' place between models where quantum particles are 'fixed' forever to their designated locations (as in [10]) and models where quantum particles move in the same space (a Bose-gas, considered in [26], [27]). In particular, this work provides a bridge between Refs [10] and [26,27]; reading this paper ahead of [26,27] might help an interested reader to get through Refs [26,27] at a much quicker pace.
We would like to note an interesting problem of analysis of the small-mass limit (cf. [15]) from the point of Mermin-Wagner phenomena.
1.3. Assumptions on the potentials. The between-sites potential V is assumed to be of class C 2 . Consequently, V and its first and second derivatives satisfy uniform bounds. Viz., 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 we need only a C 1 -smoothness): Note that for V and U (1) the bounds are imposed on their negative parts only. As to U (2) , we suppose that (a) Here ρ(x, x ′ ) stands for the (flat) Riemannian distance between points x, x ′ ∈ M. As a result of (1.3.3), there exists a 'hard core' of diameter ρ, and a given atom cannot 'hold' more than particles where v(B(ρ)) is the volume of a d-dimensional ball of diameter ρ. We will also use the bound The function J : r ∈ (0, ∞) → J(r) ≥ 0 is assumed monotonically non-increasing with r and obeying the relation J(l) → 0 as l → ∞ where Additionally, let J(r) be such that Next, we assume that the functions U (1) , U (2) and V are g-invariant: ∀ x, x ′ ∈ M and g ∈ G, In the following we will need to bound the fugacity (or activity, cf. (1.4.3)) z in terms of the other parameters of the model 1.4. The Gibbs state in a finite volume. Define the particle number operator N Λ , with the action Here, for a given x * Λ = {x * (j), j ∈ Λ}, ♯ x * Λ stands for the total number of particles in configuration x * Λ : The standard canonical variable associated with N Λ is activity z ∈ (0, ∞). give rise to positive-definite traceclass operators G Λ = G z,β,Λ and G Λ|x * We would like to stress that the full range of variables z, β > 0 is allowed here because of the hard-core condition (1.3.3): it does not allow more than κ♯Λ particles in Λ where ♯Λ stands for the number of vertices in Λ. However, while passing to the thermodynamic limit, we will need to control z and β.
Definition 1.1. We will call G Λ and G Λ|x * Γ\Λ the Gibbs operators in volume Λ, for given values of z and β (and -in the case of G Λ|x * Γ\Λ -with the boundary condition x * Γ\Λ ). The Gibbs operators in turn give rise to the Gibbs states ϕ Λ = ϕ β,z,Λ and ϕ Λ|x Γ\Λ = ϕ β,z,Λ|x Γ\Λ , at temperature β −1 and activity z in volume Λ. These are linear positive normalized functionals on the C * -algebra B Λ of bounded operators in space H Λ : and Here and below we adopt the following notational agreement: symbol ✸ marks the end of a definition, symbol ✁ the end of a statement and symbol ✷ the end of a proof.
The hard-core assumption (1.3.3) yields that the quantities Ξ(Λ) and Ξ z,β (Λ|x * Γ\Λ ) are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation. Definition 1.2. Whenever Λ 0 ⊂ Λ, the C * -algebra B Λ 0 is identified with the C * -sub-algebra in B Λ formed by the operators of the form A 0 ⊗ I Λ\Λ 0 . Consequently, the restriction ϕ Λ 0 Λ of state ϕ Λ to C * -algebra B Λ 0 is given by where Operators R Λ 0 Λ (we again call them RDMs) are positive-definite and have tr H Λ 0 R Λ 0 Λ = 1. They also satisfy the compatibility property: In a similar fashion one defines functionals ϕ Λ 0 determines a state of B Γ , see [2]. Finally, we introduce unitary operators U Λ 0 (g), g ∈ G, in H Λ 0 : Theorem 1.1. Assuming the conditions listed above, for all z, β ∈ (0, +∞) satisfying (1.3.9) and a finite Λ 0 ⊂ Γ, operators R Λ 0 Λ form a compact sequence in the trace-norm topology in H Λ 0 as Λ ր Γ. Furthermore, given any family of (finite or infinite) sets Γ = Γ(Λ) ⊆ Γ and configurations also form a compact sequence in the trace-norm topology. Any limit point, Remark. In fact, the assertion of Theorem 1.1 holds without assuming the bi-dimensionality condition on graph (Γ, E), only under an assumption that the degree of the vertices in Γ is uniformy bounded.

5.4)
Accordingly, any limiting Gibbs state ϕ of B determined by a family of limiting operators R Λ 0 obeying (1.5.4) satisfies the corresponding invariance property: ∀ finite Λ 0 ⊂ Γ, any A ∈ B Λ 0 and g ∈ G, Remarks. 1.1. Condition (1.3.9) does not imply the uniqueness of an infinite-volume Gibbs state (i.e., absence of phase transitions).
1.2. Properties (1.5.4) and (1.5.5) are trivially fulfilled for the limiting points R Λ 0 and ϕ of families {R Λ 0 Λ } and {ϕ Λ }. However, they require a proof for the limit points of the families The set of limiting Gibbs states (which is non-empty due to Theorem 1.1), is denoted by G 0 . In the Section 3 we describe a class G ⊃ G 0 of states of C * -algebra B satisfying the FK-DLR equation, similar to that in [10].

The representation for the kernels of the Gibbs operators.
A starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels ,(y,j) denotes the space of path, or trajectories, ω = ω (x,i),(y,j) in M × Γ, of time-length β, with the end-points (x, i) and (y, j). Formally, ω ∈ W β (x,i),(y,j) is defined as follows: , ω has finitely many jumps on [0, β]; if a jump occurs at time τ then d l ω, τ − , l ω, τ = 1. (2.1.1) The notation ω(τ ) and its alternative, u ω, τ , l ω, τ , for the position and the index of trajectory ω at time τ will be employed as equal in rights. We use the term the temporal section (or simply the section) of path ω at time τ . ✸ A matching (or pairing) γ between x * Λ and y * Λ is defined 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 ∈ Λ and 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 and y ∈ y * (j) there exist unique i ∈ Λ and x ∈ x * (i) such that (x, i) and (y, j) form a pair. (Owing to the condition ♯ Next, consider the Cartesian product and the disjoint union The presence of matchings in the above construction is a feature of the bosonic nature of the systems under consideration. We will work with standard sigma-algebras (generated by cylinder sets) In what follows, ξ(τ ), τ ≥ 0, stands for the Markov process on M × Γ, with cádlág trajectories, determind by the generator L acting on a function ( In the probabilistic literature, such processes are referred to as Lévy processes; cf., e.g., [21]. Pictorially, a trajectory of process ξ moves along M according to the Brownian motion with the generator −∆/2 and changes the index i ∈ Γ from time to time in accordance with jumps occurring in a Poisson process of rate In other words, while following a Brownian motion rule on M, having index i ∈ Γ and being at point x ∈ M, the moving particle experiences an urge to jump from i 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 on M from y and keeps its new index j until the next jump, and so on.
For a given pairs (x, i), (y, j) ∈ M × Γ, we denote by P β (x,i),(y,j) the nonnormalised measure on W β (x,i),(y,j) induced by ξ. That is, under measure P β (x,i),(y,j) the trajectory at time τ = 0 starts from the point x and has the initial index i while at time τ = β it is at the point y and has the index j.
where p β M (x, y) denotes the transition probability density for the Brownian motion to pass from x to y on M in time β: In view of (1.2.5), the quantity p (x,i),(y,j) and its derivatives are uniformly bounded: We suggest a term 'non-normalised Brownian bridge with jumps' for the measure but expect that a better term will be proposed in future.
Let γ be a pairing between x * Λ and y * Λ . Then P * constituting ω Λ are independent components. (Here the term independence is used in the measure-theoretical sense.) As in [10], we will work with functionals on W β x * Λ ,y * Λ ,γ representing integrals along trajectories. The first such functional, h Λ (ω Λ ), is given by Here, introducing the notation u Here h Λ (ω Λ ) is as in (2.1.10) and [10].
Finally, we introduce the indicator functional α Λ (ω Λ ): It can be derived from known results [17], [22] (for a direct argument, see [7]) that the following assertion holds true: admit the following representations: The ingredients of these representations are determined in (2. and ∀ j ∈ Γ, the number of paths ω x,i with index l x,i (τ ) = j is less than or equal to κ. Likewise, the integral in the RHS of (2.1.20) receives a non-zero contribution only come from configurations ω Λ = {ω * x,i } such that, ∀ site j ∈ Γ, the number of paths ω x,i with index l x,i (τ ) = j plus the cardinality ♯ x * (j) does not exceed κ. In particular, we will have to pass from trajectories of fixed time-length β to loops of a variable time-length. To this end, a given matching γ is decomposed into a product of cycles, and the trajectories associated with a given cycle are merged into closed paths (loops) of a time-length multiple of β. (A similar construction has been performed in [7].) To simplify the notation, we omit, wherever possible, the index β. , 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)). Accordingly, we denote by W * x,i the set W * (x,i),(x,i) . An element of W * x,i is denoted by Ω * x,i or, in short, by Ω * and called a loop at vertex i. (The upper index * indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop Ω * x,i ∈ W * x,i is denoted by k(Ω * x,i ) or k x,i . It is instructive to note that, as topological object, a given loop Ω * admits a multiple choice of the initial pair (x, i): it can be represented by 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 section at time τ ∈ [0, β] for the particle configuration {Ω * x,i (τ + βm), 0 ≤ m < k x,i } and employ the alternative notation (u(τ + βm; Ω * ), l(τ + βm; Ω * )) addressing the position and the index of Ω * at time τ + βm ∈ [0, βk(Ω * )]. ✸ Definition 2.6. Suppose x * Λ = {x * (i), i ∈ Λ} ∈ M * Λ and y * Λ = {y * (j), j ∈ Λ} ∈ M * Λ are particle configurations over Λ, with ♯ x * Λ = ♯ y * Λ . Let γ be a matching between x * Λ and y * Λ . We consider the Cartesian product and the disjoint union Accordingly, an element Ω * 3) represents a collection of paths Ω * x,i , x ∈ x * (i), i ∈ Λ, of time-length kβ, starting at (x, i) and ending up at (y, j) = γ(x, i). We will say that Ω * Λ is a path configuration in (or over) Λ.
Similarly, the RDM R Λ 0 Λ|x * Γ\Λ is determined by its integral kernel , again admitting the representation .
The focus of our interest are the numerators Ξ To introduce the appropriate representation for these quantities, we need some additional definitions.

this is not a restriction).
The functional h Λ 0 (Ω * 0 ) in Eqn (2.3.4) gives the energy of the path con- figuration Ω * 0 and is introduced similarly to Eqn (2.2.12), mutatis mutandis.

Γ\Λ
; these measures are determined by their Radon-Nikodym derivatives The first key property of the measures µ Λ and µ Λ|x * Γ\Λ is expressed in the so-called FK-DLR equation. We state it as Lemma 2.4 below; its proof repeats a standard argument used in the classical case for establishing the DLR equation in a finite volume Λ ⊂ Γ.
Given a measure µ ∈ G, we associate with it a normalized linear functional ϕ = ϕ µ on the quasilocal C * -algebra B. First, we set This defines a kernel F Λ 0 (x * 0 , y * 0 ), x * 0 , y * 0 ∈ M * Λ 0 , where Λ 0 ⊂ Γ is a finite set of sites. It is worth reminding the reader of the presence of the indicator functionals 1 · ∈ F Λ 0 in (3.1.5) and (3.1.6) (in the integral . These indicators guarantee the compatibility property: ∀ finite Λ 0 ⊂ Λ 1 , when Λ ր Γ. Then we use Lemma 1.1 from [10] to derive that the sequence of the RDMs R Λ 0 Λ and R Λ 0 Λ|x * Γ\Λ is compact in the trace-norm operator topology To verify compactness of the RDMKs F Λ 0 Λ (x * 0 , y * 0 ) and F Λ 0 we, again as in [10], use the Ascoli-Arzela theorem, which requires the properties of uniform boundedness and equicontinuity. These properties follow from Lemma 4.1. (i) Under condition (1.3.9) the RDMKs F Λ 0 Λ (x * 0 , y * 0 ) and F Λ 0 The first contribution can again be uniformly bounded in terms of the constant p M . The detailed argument, as in [10], includes a deformation of a trajectory and is done similarly to [10] (the presence of jumps does not change the argument because p M yields a uniform bound in (2.1.7)).
The second contribution yields, again as in [10], an expression of the form W * x * 0 ,y * 0 P * where the functional h x,i (Ω * (x,i),γ 0 (x,i) , Ω * Λ\Λ 0 ) is uniformly bounded. Combining this an upper bound similar to (4.1.2) yields the desired estimate for the gradients in (4.1.3). ✷ Hence, we can guarantee that the RDMs R Λ 0 Λ and R Λ 0 Λ|x * Γ\Λ converge to a limiting RDM R Λ 0 along a subsequence in Λ ր Γ. The diagonal process yields convergence for every finite Λ 0 ⊂ Γ. A parallel argument leads to compactness of the measures µ Λ 0 Λ for any given Λ 0 as Λ ր Γ. We only give here a sketch of the corresponding argument, stressing differences with its counterpart in [10].
In the probabilistic terminology, measures µ Λ represent random marked point fields on M × Γ with marks from the space W * = W * 0,0 where W * 0 = k≥1 W kβ 0 and W kβ 0 is the space of loops of time-length kβ starting and finishing at 0 ∈ M and exhibiting jumps, i.e., changes of the index. (The space W * x,i introduced in Definitions 2.1 and 2.5 can be considered as a copy of W * placed at site i ∈ Γ and point x ∈ M.) The measure µ Λ 0 Λ describes the restriction of µ Λ to volume Λ 0 (i.e., to the sigma-algebra W * Λ 0 ) and is given by its Radon-Nikodym derivative p Λ 0 Λ relative to the reference measure dΩ * Λ 0 on W * Λ 0 (cf. (2.2.11), (2.4.3)). The reference measure is sigma-finite. Moreover, under the condition (1.3.9), the value p Λ 0 Λ (Ω * Λ 0 ) is uniformly bounded (in both Λ ր Γ and Ω * Λ 0 ∈ W * Λ 0 ). This enables us to verify tightness of the family of measures {µ Λ 0 Λ , Λ ր Γ} and apply the Prokhorov theorem. Next, we use the compatibility property of the limit-point measures µ Λ 0 Γ and apply the Kolmogorov theorem. This establishes the existence of the limit-point measure µ Γ .
The vectors corresponding to g for j ∈ Λ(n), or even for j ∈ Γ, as it agrees with the requirement that g (n) j ≡ g when j ∈ Λ 0 and g (n) j ≡ e for j ∈ Γ \ Λ(n).
Accordingly, we will use the notation g Λ(n) = {g (n) j , j ∈ Λ(n)}. Observe that the tuned family g Λ(n)\Λ 0 does not change the contribution into the energy functional h Λ(n)|Γ\Λ(n) coming from potentials U ('1) and U (2) : it affects only contributions from potential V .
The Taylor formula for function V , together with the above identification of vectors θ (n) j , gives: ≤ C |θ| 2 |υ(n, j) − υ(n, j ′ )| 2 V , x, x ′ ∈ M. Here C ∈ (0, ∞) is a constant, |θ| stands for the norm of the vector θ representing the element g and the value V is taken from (1.3.1).
Next, the square |υ(n, j) − υ(n, j ′ )| 2 can be specified as  Therefore, it remains to estimate the sum