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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 637375, 20 pages
A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model
1Swansea University, Singleton Park, Swansea SA2 8PP, UK
2Instituto de Mathematica e Estatistica, USP, Rua de Matão, 1010, Cidada Universitária, 05508-090 São Paulo, SP, Brazil
3Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
4IITP, RAS, Bolshoy Karetny per. 18, Moscow 127994, Russia
Received 20 March 2013; Accepted 14 May 2013
Academic Editor: Christian Maes
Copyright © 2013 Mark Kelbert and Yurii Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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. In the model considered here the phase space of a single spin is where is a -dimensional unit torus with a flat metric. The phase space of spins is , the subspace of formed by functions symmetric under the permutations of the arguments. The Fock space yields the phase space of a system of a varying (but finite) number of particles. We associate a space with each vertex of a graph satisfying a special bidimensionality property. (Physically, vertex 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) , the minus a half of the Laplace operator on , 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 , , describing a field generated by a heavy atom, (b) two-body potentials , , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials , , scaled along the graph distance between vertices , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group acts on , represented by a Euclidean space or torus of dimension , preserving the metric and the volume in . Furthermore, we suppose that the potentials , , and are -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is -invariant, provided that the thermodynamic variables (the fugacity and the inverse temperature ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.
1.1. Basic Facts on Bi-Dimensional Graphs
As in , we suppose that the graph has been given, with the set of vertices and the set of edges . The graph has the property that whenever edge , the reversed edge belongs to as well. Furthermore, graph is without multiple edges and has a bounded degree; that is, the number of edges with a fixed initial or terminal vertex is uniformly bounded: The bi-dimensionality property is expressed in the bound where stands for the set of vertices in at the graph distance from (a sphere of radius around ): (The graph distance between is determined as the minimal length of a path on joining and .) This implies that for any the cardinality of the ball grows at most quadratically with .
A justification for putting a quantum system on a graph can be that graph-like structures become increasingly popular in rigorous Statistical Mechanics, for example, in quantum gravity. Namely, see [2–4]. On the other hand, a number of properties of Gibbs ensembles do not depend upon “regularity” of an underlying spatial geometry.
1.2. A Bosonic Model in the Fock Space
With each vertex we associate a copy of a compact manifold which we take in this paper to be a unit -dimensional torus with a flat metric and the volume . We also associate with a bosonic Fock-Hilbert space . Here where is the subspace in formed by functions symmetric under a permutation of the variables. Given a finite set , we set . An element is a complex function: Here is a collection of finite point sets associated with sites . Following , we call a particle configuration at site (which can be empty) and a particle configuration in, or over, . The space of particle configurations in can be represented as the Cartesian product where is the disjoint union and is the collection of (unordered) -point subsets of . (One can consider as the factor of the “off-diagonal” set in the Cartesian power under the equivalence relation induced by the permutation group of order .) The norm and the scalar product in are given by where measure is the product and is the Poissonian sum measure on : Here is the volume of torus .
As in , we assume that an action is given, of a group that is a Euclidean space or a torus of dimension . The action is written as Here vector with components and is the -dimensional vector representing the element , where is a matrix of column rank with rational entries. The action of is lifted to unitary operators in : where and .
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 . In our context the Hubbard Hamiltonian of the system in acts as follows: Here means the Laplacian in variable . Next, stands for the cardinality of the particle configuration (i.e., when ), and the parameter is introduced in (17). (Symbol will be used for denoting the cardinality of a general (finite) set; for example, means the number of vertices in .) Further, denotes the particle configuration with the point removed and point added to .
As in , we also consider a Hamiltonian in an external field generated by a configuration where is a (finite or infinite) collection of vertices. More precisely, we only consider with (see (17) below) and set
We will suppose that vanishes if the graph distance . We will also assume uniform boundedness: in view of (1) it implies that the total exit rate from site 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  (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 is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials , and 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 is 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 is interesting from the point of view of transport phenomena that they may display. (An analogy with the famous Anderson model, in its multiparticle version, inevitably comes to mind; cf., e.g., .) Methodologically, such systems occupy an “intermediate” place between models where quantum particles are “fixed” forever to their designated locations (as in ) and models where quantum particles move in the same space (a Bose-gas, considered in [8, 9]). In particular, this work provides a bridge between [1, 8, 9]; reading this paper ahead 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 analysis of the small-mass limit (cf. ) from the point of Mermin-Wagner phenomena.
1.3. Assumptions on the Potentials
The between-sites potential is assumed to be of class . Consequently, and its first and second derivatives satisfy uniform bounds. Namely, Here and run through the pairs of variables , . A similar property is assumed for the on-site potential (here we need only a smoothness): Note that for and the bounds are imposed on their negative parts only.
As to , we suppose that (a) and (b) a -function such that whenever . Here stands for the (flat) Riemannian distance between points . As a result of (16), there exists a “hard core” of diameter , and a given atom cannot “hold” more than particles where is the volume of a -dimensional ball of diameter . We will also use the bound Formally, (16) means that the operators in (11) and (12) are considered for functions vanishing when in the particle configuration , the cardinality for some .
The function is assumed monotonically nonincreasing with and obeying the relation as , where Additionally, let be such that Next, we assume that the functions , , and are -invariant: and , In the following we will need to bound the fugacity (or activity, cf. (25)) in terms of the other parameters of the model
1.4. The Gibbs State in a Finite Volume
Define the particle number operator , with the action Here, for a given , stands for the total number of particles in configuration : The standard canonical variable associated with is activity .
The Hamiltonians (11) and (12) are self-adjoint (on the natural domains) in . Moreover, they are positive definite and have a discrete spectrum, cf. . Furthermore, , and give rise to positive-definite trace-class operators and : We would like to stress that the full range of variables is allowed here because of the hard-core condition (16): it does not allow more than particles in where stands for the number of vertices in . However, while passing to the thermodynamic limit, we will need to control and .
Definition 1. We will call and the Gibbs operators in volume , for given values of and (and—in the case of —with the boundary condition ).
The Gibbs operators in turn give rise to the Gibbs states and , at temperature and activity in volume . These are linear positive normalized functionals on the -algebra of bounded operators in space : where
The hard-core assumption (16) yields that the quantities and are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation.
Definition 2. Whenever , the -algebra is identified with the subalgebra in formed by the operators of the form . Consequently, the restriction of state to -algebra is given by where Operators (we again call them RDMs) are positive definite and have . They also satisfy the compatibility property: , In a similar fashion one defines functionals and operators , with the same properties.
1.5. Limiting Gibbs States
The concept of a limiting Gibbs state is related to notion of a quasilocal -algebra; see . For the class of systems under consideration, the construction of the quasilocal -algebra is done along the same lines as in : is the norm completion of the algebra . Any family of positive-definite operators in spaces of trace one, where runs through finite subsets of , with the compatibility property determines a state of , see [12, 13].
Finally, we introduce unitary operators , , in : where
Theorem 3. Assuming the conditions listed above, for all satisfying (22) and a finite , operators form a compact sequence in the trace-norm topology in as . Furthermore, given any family of (finite or infinite) sets and configurations with , operators also form a compact sequence in the trace-norm topology. Any limit point, , for or as , is a positive-definite operator in of trace one. Moreover, if and and are the limits for and or for and along the same subsequence , then the property (32) holds true.
Consequently, the Gibbs states and form compact sequences as .
Remark 4. In fact, the assertion of Theorem 3 holds without assuming the bidimensionality condition on graph , only under an assumption that the degree of the vertices in is uniformy bounded.
Definition 5. Any limit point for states and is called a limiting Gibbs state (for given ).
Theorem 6. Under the condition (22), any limiting point, , for or , as , is a positive-definite operator of trace one commuting with operators : , Accordingly, any limiting Gibbs state of determined by a family of limiting operators obeying (35) satisfies the corresponding invariance property: finite , any , and ,
Remarks. (1) Condition (22) does not imply the uniqueness of an infinite-volume Gibbs state (i.e., absence of phase transitions).
2. Feynman-Kac Representations for the RDM Kernels in a Finite Volume
2.1. 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 and of operators and .
Definition 7. Given , denotes the space of path, or trajectories, in , of time-length , with the end-points and . Formally, is defined as follows: The notation and its alternative, , for the position and the index of trajectory at time will be employed as equal in rights. We use the term the temporal section (or simply the section) of path at time .
Definition 8. Let , and be particle configurations over , with . A matching (or pairing) between and is defined as a collection of pairs , with , , and , with the properties that (i) and there exist unique and such that and form a pair, and (ii) and there exist unique and such that and form a pair. (Owing to the condition , these properties are equivalent.) It is convenient to write .
Next, consider the Cartesian product and the disjoint union Accordingly, an element in (38) represents a collection of paths , , , of time-length , starting at and ending up at . We say that is a path configuration in (or over) .
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 , , and .
Definition 9. In what follows, , , stands for the Markov process on , with cádlág trajectories, determined by the generator acting on a function by In the probabilistic literature, such processes are referred to as Lévy processes; see, for example, .
Pictorially, a trajectory of process moves along according to the Brownian motion with the generator and changes the index 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 , having index and being at point , the moving particle experiences an urge to jump from to a neighboring vertex and to a point at rate . After a jump, the particle continues the Brownian motion on from and keeps its new index until the next jump, and so on.
For a given pairs , we denote by the nonnormalised measure on induced by . That is, under measure the trajectory at time starts from the point and has the initial index while at time it is at the point and has the index . The value is given by where denotes the transition probability density for the Brownian motion to pass from to on in time : In view of (13), the quantity and its derivatives are uniformly bounded: where is a constant.
We suggest a term “non-normalised Brownian bridge with jumps” for the measure but expect that a better term will be proposed in future.
Definition 10. Suppose that and are particle configurations over , with . Let be a pairing between and . Then denotes the product measure on : Furthermore, stands for the sum measure on :
According to Definition 10, under the measure , the trajectories constituting are independent components. (Here the term independence is used in the measure-theoretical sense.)
As in , we will work with functionals on representing integrals along trajectories. The first such functional, , is given by Here, introducing the notation and for the positions in of paths and at time , we define Next, with and standing for the indices of and at time ,
Next, consider the functional : for . As before, we assume that . Define Here is as in (46) and where, in turn, The functionals and are interpreted as energies of path configurations. Compare and in .
Lemma 11. For all and a finite , the Gibbs operators and act as integral operators in : Moreover, the integral kernels and vanish if . On the other hand, when , the kernels and admit the following representations: The ingredients of these representations are determined in (46)–(51).
Remark 12. As before, we stress that, owing to (16) and (17), a nonzero contribution to the integral in the RHS of (54) can only come from a path configuration such that and , the number of paths with index is less than or equal to . Likewise, the integral in the RHS of (55) receives a non-zero contribution only from configurations such that, site , the number of paths with index plus the cardinality does not exceed .
2.2. The Representation for the Partition Function
The FK representations of the partition functions in (27) and in (1.4.6) reflect a specific character of the traces and in . The source of a complication here is the jump terms in the Hamiltonians and in (11) and (12), respectively. In particular, we will have to pass from trajectories of fixed time-length to loops of a variable time length. To this end, a given matching 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 . (A similar construction has been performed in .)
To simplify the notation, we omit, wherever possible, the index .
Definition 13. For given , the symbol denotes the disjoint union:
In other words, is the space of paths in , of a variable time-length , where takes values and called the length multiplicity, with the end-points and . The formal definition follows the same line as in (37), and we again use the notation and the notation for the pair of the position and the index of path at time . Next, we call the particle configuration the temporal section (or simply the section) of at time . We also call a path (from to ).
A particular role will be played by closed paths (loops), with coinciding endpoints (where ). Accordingly, we denote by the set . An element of is denoted by or, in short, by and called a loop at vertex . (The upper index indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop is denoted by or . It is instructive to note that, as topological object, a given loop admits a multiple choice of the initial pair : it can be represented by any pair at a time where . As above, we use the term the temporal section at time for the particle configuration and employ the alternative notation addressing the position and the index of at time .
Definition 14. Suppose and are particle configurations over , with . Let be a matching between and . We consider the Cartesian product:
and the disjoint union:
Accordingly, an element in (58) represents a collection of paths , , , of time-length , starting at and ending up at . We say that is a path configuration in (or over) .
Again, loops play a special role and deserve a particular notation. Namely, denotes the Cartesian product: and stands for the disjoint union (or equivalently, the Cartesian power):
Denote by a collection of loop configurations at vertices starting and ending up at particle configurations (note that some of the ’s may be empty). The temporal section (or, in short, the section), , of at time is defined as the particle configuration formed by the points where , , and .
As before, consider the standard sigma algebras of subsets in the spaces , , , , , and introduced in Definitions 13 and 14. In particular, the sigma algebra of subsets in will be denoted by ; we comment on some of its specific properties in Section 3.1. (An infinite-volume version of is treated in Section 3.2 and after.)
Definition 15. Given points , we denote by the sum measure on :
Further, denotes the similar measure on :
Definition 16. Let and be particle configurations over , with . Let be a matching between and , and we define the product measure :
and the sum measure
Next, symbol stands for the product measure on :
Finally, yields the measure on : Here, for , we set: . For sites with , the corresponding factors are trivial measures sitting on the empty configurations.
We again need to introduce energy-type functionals represented by integrals along loops. More precisely, we define the functionals and which are modifications of the above functionals and ; confer (46) and (50). Say, for a loop configuration over with an initial and final particle configuration , To determine the functionals and , we set, for given and : A (slightly) shortened notation is used for the index and for the position for , of the section of the loop at time , and similarly with and . (Note that the pairs and may coincide.) Then
Next, the functional takes into account the bosonic character of the model: The factor in (70) reflects the fact that the starting point of a loop may be selected among points arbitrarily.
Next, we define the functional : for