Contractions of algebras underlying models for superfluidity

Not unitary transformation based on algebraic contraction is proposed to realize mappings between models Hamiltonians. Such a transformation contracts the algebra of the degrees of freedom underlying the Hamiltonian. The rigorous mapping between the anisotropic XXZ Heisenberg model, the Quantum Phase Model, and the Bose Hubbard Model is identified as the group contractions of the algebra u(2) underlying the dynamics of the XXZ Heisenberg model. PACS numbers: 05.70.Fh, 68.35.Rh, 02.20.-a, 74.50.+r The problem of mapping between not equivalent algebras was solved, in mathematical physics, years ago by Inönü and Wigner [1] and subsequently generalized by Saletan [2] when they founded the concept of contraction of a Lie algebra A [3]. Algebraic contraction is a transformation which may be singular on A’s basis (namely, the kernel of the transformation is non trivial), while it is regular on its commutation brackets [4]. Further application of algebraic contractions in physics traces back to studies of Umezawa and coworkers [5]. They shown, under quite general hypothesis, that in a zero temperature phase transition, the symmetry of the system in the disordered phase (is rearranged) contracts (through contraction of the algebra spanned by degrees of freedom of the system), onto the symmetry of the ordered phase. For example, in Heisenberg ferromagnets the broken symmetry so(3) (which is the spin algebra and which accounts for the rotation symmetry of the magnetization in the paramagnetic phase) is contracted onto the euclidean symmetry e(2) of the traslators (which accounts for the reduced symmetry rotation of the system around the magnetization axes) [5]. The idea developed in the present work consists in considering contractions of algebras spanned by the degrees of freedom of the system as establishing a link between models which are intrinsically distinct (in the sense they are not unitarly connectible). I will furnish an application of such idea in condensed matter physics: I will show that contractions can provide exact mapping between the Bose Hubbard model, the quantum Josephson model and certain anisotropic Heisenberg model. The motivation is to found rigorously the relation between these three models, which is employed to describe low temperature behaviour in various mesoscopic systems. The Bose Hubbard Model (BHM) describes a lattice gas of interacting charged bosons [6]. It is related to Quantum Phase Model (QPM) which is largely employed in the physics Contractions of algebras underlying models for superfluidity 2 of Josephson junctions arrays since it can describe the competition between quantum phase coherence and Coulomb blockade[7]. The elementary degrees of freedom entering the QPM are the phases of the superconducting order parameter φj and the charge unbalance to charge neutrality Nj := −i∂φj (its eigenvalues range in (−∞,+∞)) in the island j. These two variables are considered as canonically conjugated in the QPM. The phase diagrams of BHM and QPM were analyzed by many authors [8]. They describe zero temperature quantum phase transitions between incompressible insulators and coherent superfluid phases. Finally, theXXZ anisotropic Heisenberg model [9] shows a low temperature behaviour related to those ones of BHM and QPM. In particular, its zero temperature phase diagram shows phase transitions from paramagnetic to canted phases that can be interpreted as insulator to superfluid phase transition [10]. Up to the present study, the relation between the BHM, the QPM and the XXZ model consisted in the fact that they belong to the same universality class. Unitary transformations mapping one model on each other do not exists. In fact, the arguments usually employed to relate such models on each other did not want to be rigorous. For instance, the phase–number variables entering the QPM cannot be thought as mathematically originated from bosonic operators in BHM since a no–go theorem forbids aj ∼ √njeiφj , a†j ∼ ej √ nj (even with the widest reasonable latitude of interpretation [11, 12]) as long as the phases φj are hermitian and canonically conjugated to a bounded ni (as it is the bosonic number operator). A way out from this difficulty is realized in QPM by removing the hypothesis of boundness from below of ni. It is worthwhile noting that connections between ni and Ni cannot be unitary since unitary transformations cannot transform bounded into unbounded operators. In contrast, algebraic contractions can connect them. I will use this variation of algebras’ “topology” as the crucial tool to realize the mapping between the three models I deal with. Such a transformation induces also the mapping of the matrix elements of the Hamiltonians. The paper is organized as follow. After having outlined the general procedure accounting of contracting the underlying algebra characterizing quite general Hamiltonians, then it is applied to mapping between the BHM, the QPM, and the XXZ. Such a mapping, gives algebraic foundations to the procedure developed in the ref. [13] to map the zero temperature phase diagram of the BHM onto the phase diagrams of the QPM and of the XXZ model within (suitable) mean field approximation. I assume models Hamiltonian on a lattice Λ writable in terms of generators of a given Lie algebra A = ⊕i∈Λ gi having the form: